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Experiment No.1 PHYSICAL PENDULUMExperiment No.11 HOOKE’S LAW; DETERMINATION OF YOUNG’S MODULUSExperiment No.13 VISCOSITY COEFFICIENTExperiment No.22 VACUUM. BOYLE-MARIOTTE’s LAWExperiment No.25 INTERFERENCE OF ACOUSTIC WAVESExperiment No.33 DIELECTRIC CONSTANTExperiment No.41 MEASURING EARTH’S MAGNETIC FIELDExperiment No.51 INDEX OF REFRACTION FOR SOLIDSExperiment No.52 INDEX OF REFRACTION FOR LIQUIDSExperiment No.54 GLASS PRISM; CHROMATIC DISPERSIONExperiment No.61 ELECTROMAGNETIC OSCILLATIONS IN AN RLC CIRCUITExperiment No.71 DIFFRACTION AND INTERFERENCE OF LASER LIGHTExperiment No.92 ABSORPTION OF GAMMA RAYS

compiled by dr hab. Andrzej Lendaon the basis of: ”PRACOWNIA FIZYCZNA” WFiTJ, Andrzej Ziba (editor), skrypt AGH 1529,

The black-and-white figures come from the mentioned text-book (with permission of the Ed-itor); the colour plates are reproduced from the: David Halliday, Robert Resnick and JearlWalker: Fundamentals of Physics. This excellent book is the No.1 reference for the students ofthe International School of Technology. Several copies of this book have been bought by theIST and are available in the School’s library.

revised edition, Cracow, October 2000

PHYSICS LABORATORY — page 1-1

Experiment No.1 — PHYSICAL PENDULUM

Purpose:Observations of the oscillatory motion of a physical pendulum. Determination of the momentof inertia of rigid bodies (simple geometrical shapes).

Introduction:A mathematical pendulum consists of a point-like mass m attached to a weightless cord ofa length l. A physical pendulum (cf. Fig.1-1) is a rigid body, which can rotate around anaxis 0. The axis 0 is at a distance l from the center of mass of the body C. In the limit of anegligible size of the body a physical pendulum becomes a mathematical one.If deflected laterally from the equilibrium position (θ = 0) through an angle θ, the pendu-lum performs the oscillatory motion. The equation governing the motion is the 2nd Law ofDynamics, i.e.:

I0d2θ

dt2= −mg l sin θ, (1.1)

where I0 is the pendulum rotational inertia, d2θ/dt2 = ε is the pendulum’s angular acceleration,and mgl sin θ is the absolute value of the restoring torque.For a small initial deflection, sin θ ≈ θ and equation 1.1 becomes

d2θ

dt2+ ω2

0θ = 0, (1.2)

with ω20

def=

mgl

I0

. The solution of Eq. 1.2 is the usual

θ(t) = θmax cos(ω0t + α), (1.3)

with constants θmax and α that can be determined from the initial conditions. The solutionrepresents the simple harmonic motion with the period, T , equal to

T = 2π

√I0

mgl. (1.4)

The experiment consists in measuring the period T , the axis-center-of-mass distance l and themass m. Once these quantities are known equation 1.4 can be solved for I0. Note that thisquantity is the rotational inertia about an axis of rotation that passes through the point ofsuspension, 0. The inertia about an axis of rotation that passes through the center-of-mass ,IC , can be calculated from the parallel-axis theorem:

I0 = IC + ml2. (1.5)

It is worth noting, that in the limit of a point-like mass IC = 0, so I0 = ml2, and insertingthis value into equation 1.4 we recover the well known formula for the oscillation period of themathematical pendulum.


Figure 1-1: A physical pendulum.

For simple geometrical shapes (like those used in the experiment) the rotational inertia can becalculated analytically, using the basic definition of the inertia that describes this quantity asan integral of all the point-like mass-particles dm multiplied by the squares of their distancesfrom the axis of rotation, r2:

I =∫

mr2dm (1.6)

with the integration carried out over the whole volume of the body. For a rod (mass m, lengthL) and for a ring (mass m, inner radius Rw, outer radius Rz) the IC values are (cf. Fig.1-2 –the rotation axis is perpendicular to the figure plane):

IrodC =

m

12L2, I

ringC =

m

2

(R2

w + R2z

). (1.7)

Thus, the values of IC measured experimentally (via the measurement of T ) can be confrontedwith the values calculated accordingly to equation 1.7. This requires experimental determina-tion of mass and geometrical parameters of the pendula, usually with the help of simple tools.


Figure 1-2: Rod and ring and their parameters to-be-measured.

Experimental SetupThe experimental setup consist of a rod and ring, which can be suspended on an appropriatestand. The measuring devices are: simple weighing scale (pan scales), slide caliper, measuringtape and timer.

A. Determination of inertia for ring and rod:

Experimental Procedure

1. Weigh the rod and the ring, using the pan scales.

2. Measure the lengths: a, l and radii Rw, Rz using the measuring tape and slide caliper.

3. After hanging the rod (ring) on the stand, make it oscillate (the initial deflection shouldbe no greater than few degrees) and measure the time of 30-40 oscillations. Repeat thismeasurement 10 times and calculate the period T as an arithmetic mean.

Report

1. Calculate (using equations 1.4 and 1.5) the rotational inertia: I0 and IC for the ring andfor the rod.

2. Calculate the errors of determination.

3. Calculate the corresponding IC values and their errors using the formulae 1.7.

4. Compare the IC values obtained using these two methods.


B. Determination of the probability distribution function for the measured T values:


1. Measure 100 times the time interval τ of two (three) full oscillations. Use a precise timer(0.01 s precision). The initial amplitude should be small; carry out several measurementswith the pendulum still swinging.

Report

1. Split the range of the results (τ values) into n ≈ 10 intervals of an equal width ∆τ .Classify the measured values, calculate the relative frequencies ni/n (i = 1, 2, . . . , n).

2. Construct the frequency histogram.

3. Calculate the mean value τ and the sample mean square deviation Sτ . Standardize yourhistogram τ variable:

U =τ − τ

Sτ

.

Standardize also the frequencies – replace the actual frequencies with the relative ones(the sum of all the histogram bars must be equal to unity.

4. Redraw the histogram inserting the standard normal distribution.

5. Test the hypothesis about the normal distribution of the τ values using the Pearson χ2

test.

Questions for discussion:

• Define the physical quantities that describe the rotational motion (kinetic: angle, angularvelocity and acceleration; dynamic: angular momentum, rotational inertia, torque).

• Second law of dynamics for rotation: torque = ddt

(angular momentum).

• Rotational inertia I: definition. Calculation of I for a homogeneous rod (length l, massM) rotating around the axis perpendicular to the rod and passing through its center ofmass. Parallel axis theorem (Steiner’s theorem).

• Harmonic force and harmonic motion. Physical pendulum. Motion of physical pendulumas an example (simplified!) of the harmonic motion.


Experiment No.11HOOKE’S LAW; DETERMINATION OF YOUNG’S MODULUS

Purpose:Determination of Young modulus for various materials via the static method.

Introduction:The concept of ”rigid body” used in the dynamics of rotational motion is in reality an ab-straction. It assumes that the forces acting on a body do not change its shape (volume). Inreal situations, applied forces do change the body’s actual shape, although relative changes odbody’s dimensions are usually very small and, therefore, in many cases may be neglected. Wechange body’s dimensions by pulling, pushing, twisting, or compressing it. To get a feelingfor the orders of magnitude involved consider a steel rod, 1 meter long and 1 centimeter indiameter. If you hang a subcompact car from the end of such a rod, the rod will stretch, butonly about 0.5 millimeter (0.05%). Furthermore, the rod will return to its original length whenthe car is removed.If the shape returns to the initial one once the deforming forces have been removed, we definethe deformations elastic. The Hooke’s law, formulated as far back as in the 17th century, statesthat the degree of deformation is directly proportional to the magnitude of the deformationforces.In order to analyze the phenomenon quantitatively, let us consider a uniform rod (cf. Fig.11-1).The increase of the rod’s length, ∆L, is proportional to the actual rod’s length L and actingforce F , and inversely proportional to the area of the rod’s cross-section A. The coefficientof proportionality in this relation 1/E depends on the type of material and the quantity E iscalled Young modulus:

∆L =FL

EA. (11.8)

A more compact version of the Hooke’s law may be obtained if one introduces the concept ofrelative elongation or strain ε = ∆L/L as a measure of the body’s deformation and stressσ = F/A as a measure of strength of the deforming force. Both deformation and strainare essentially vector quantities – the scalar form of the relation assumes tacitly that bothvectors (elongation and deforming force) are perpendicular to the transverse cross-section ofthe deformed body. Such a stretching (or compressing) stress is called tensile stress and wehave:

σ = Eε. (11.9)

It should be strongly emphasized that the Hooke’s law applies only to a limited range of possi-ble deformations. The relationship strain-stress (cf. Fig.11-2) is linear for a substantial rangeof applied stresses. Then the linearity begins to wane and if the stress is increased beyondthe yield strength of the specimen (Fig.11-2), the specimen becomes permanently deformedand does not recover its original shape (dimensions) when the stress is removed. If the stresscontinues to increase, the specimen eventually ruptures, at the stress called ultimate stress.


Figure 11-1: A cylinder subject to tensile stress stretches by an amount ∆L.

Figure 11-2: A typical strain-stress relationship.

Experimental SetupThe experimental setup is a simple one: a special stand including a micrometric strain gauge,weights, a micrometric screw and a rule.The method of E determination uses directly equation 11.8. In a special stand, portions of wiremade of such materials as Fe, Al, Cu, etc. are hung by the upper end, whereas the lower end isconnected to a plate, which gradually is weighted down. The elongation of the wire is measuredwith the aid of a micrometric strain gauge. Because of the construction of the whole system(cf. Fig.11-3) the readings of the gauge correspond to 200% of the actual increase in length.The relationship obtained ∆L = ∆L(m), where m is the weight’s mass, should be a linear onewith the straight line fitted to the experimental points having a slope a — ∆L = a ·m. Then


the looked-for E will be given by:

E =gL

aA. (11.10)


Figure 11-3: Experimental setup for measuring Young modulus (static method).

1. Verify the vertical positioning of the stand, using the levelling screws at the stand’s legsand observing the plumb-line position.

2. Measure the lengths of the steel wire, using the rule and the wires’ diameters with the aidof micrometric screw. The measurement of the diameter has to be carried out 10-15 times,in various points of the wire’s length (an arithmetic mean should be used in calculations)

3. Measure the elongation of the wire under the action of different weights. The weightsshould be increased from 1 to 10 kilograms with a step of 1 kg. Repeat the measurementof the wire elongation with the weights varied in a decreasing order.

4. Repeat the whole measurement for a copper or brass wire.

Report (for both wires)

1. Construct the graph: ∆L = ∆L(m).

2. Fit a straight line to the experimental points. Find the slope value a and its error ∆a. (N.B. If the first points show a poor linearity reject them – they correspond to a preliminary”straightening” of the wire.)

3. Calculate Young modulus E, using the a value (from the fit) and measured L and A.Calculate the error of E determination. Confront your results with the data from physicaltables



• Elastic deformation and Hooke’s law.

• Young modulus. Definition, units.

• Linear regression (statistics!) as a tool for constructing a linear relationship from mea-surement data.


Experiment No.13 — VISCOSITY COEFFICIENT

Purpose:Determination of the viscosity coefficient of a real liquid using the Stokes’ method (fallingsphere).

Introduction:When studying the real fluid (liquid or gas) flow one observes some frictional phenomena (energydissipation in a volume of liquid). This phenomena occur in the whole volume of fluid and bearthe name of intrinsic friction or viscosity.In order to visualize this phenomenon, let us consider two plates immersed in a viscous liquid(or gas). If one plate is moving with respect to the other one, the frictional braking force (tobe overcome by an external applied force, if the motion is to be continued) F is proportional tothe area S of each of the plates, and to the relative velocity v and inversely proportional to theinter-plate distance d. The coefficient of the proportionality is the viscosity coefficient η, amaterial constant that depends on the nature of liquid (or gas) and temperature, whose unitsare Pa·s. The equation relating all the introduced quantities takes the form (cf. Fig.13-1):

F = ηS v

d. (13.11)

Figure 13-1: Motion of two plates in a viscous fluid.assumed distribution of the fluid velocities

Fall of a sphere in a viscous liquidConsider a sphere falling under the action of gravity in a viscous liquid. If the velocity of thefall is relatively small, the motion can be treated as a laminar one, i.e. no turbulences orvortices accompany the sphere’s motion. Besides the gravity, the forces acting on the sphereare: the hydrostatic lift Fw and the viscosity force (viscosity drag) Fo. The latter can be shownto be equal to

Fo = 6πηrv, (13.12)


where v is the velocity of the falling sphere and r is its radius. The above formula (the so-calledStokes’ formula) applies if the sphere moves through an unlimited volume of liquid. For asphere falling in a cylinder (along its axis) of the radius R the above formula has to be slightlymodified and it becomes

Fo = 6πηrv(1 + 2.4

r

R

)≡ K v, (13.13)

with K = 6πηr(1 + 2.4r/R).The hydrostatic force being equal to the product of the sphere volume V , liquid’s density ρ andgravity g, the equation of motion (Cf. Fig.13-2) for the sphere’s velocity becomes:

mdv

dt= mg − ρgV −Kv. (13.14)

Note, that the two forces (mg and ρgV ) are constant terms. This equation can be readily

Figure 13-2: Sphere falling in a viscous fluid.

solved for the velocity v:

v(t) = veq + (v0 − veq)e−tK/m ≡ veq + (v0 − veq)e

−t/τ . (13.15)

The time constant τ is equal to the ratio m/K and may be perceived as a ”natural” time-unit;v0 is the initial sphere’s velocity, at t = 0, and veq is the velocity attained by a sphere for whichthe gravity force and hydrostatic lift difference becomes exactly balanced by the viscosity drag:

Kveq = mg − ρgV. (13.16)

As the exponential term in 13.15 becomes negligible after t ≈ 3τ the observed sphere’s motioncan be approximated as a uniform one with the velocity

v(t) = veq =(m− ρV )g

6πηr(1 + 2.4 r

R

) . (13.17)


A simple transformation of equation 13.7 allows to calculate η if the remaining quantities areknown (measured). The veq velocity can be measured as the quotient of the distance l (cf.Fig.13-4) traveled by the sphere (once the uniform motion has set in) during the time T :

veq =l

T. (13.18)

A graph of equation 13.7 is shown in Fig.13-3.

Figure 13-3: Sphere’s velocity versus time.


Experimental SetupThe experimental setup (cf. Fig.13-4) consist of a glass cylinder filled by a viscous liquid(glycerin,CH2OH·CHOH·CH2OH). Two horizontal bars determine the l distance that has to be used formeasuring veq.

Figure 13-4: Experimentalsetup for measuring veq.

Experimental

1. Weigh the balls using either an analytical or torsion balance.

2. Measure the ball diameters 2r using a micrometer screw.

3. Measure the l distance and the cylinder inner diameter 2R.

4. Read out the temperature of the liquid.

5. Measure the time T of the fall for all spheres.

Report

1. Calculate (using equation 13.17) the viscosity coefficient, η.

2. Calculate the error of the η determination.

3. Calculate the values of the Reynolds number for all the spheresused.Reynolds number Re is given by the equation

Re =vLρ

η, (13.19)

where the notation is as above. L is a linear dimension of the bodyperpendicular to direction of the motion. For a sphere L = 2r.


• Viscosity; viscosity coefficients (units!)

• Archimedes Law (hydrostatic lift). Why some bodies float and others sink?

• Forces acting on a ball moving under the gravity through a viscous liquid. Equation ofmotion. Its solution (for the velocity).


Experiment No.22 — VACUUM. BOYLE-MARIOTTE’s LAW

Purpose:Examination of a standard physical setup for obtaining a low-pressure vacuum. Verification ofthe Boyle-Mariotte law for the room temperature. Determination of the gas constant R.

Introduction:Under the term vacuum we understand a space filled with gases or vapours of a pressure thatis significantly lower than the atmospheric one. Quantitatively, the vacuum – or vacuum level– can be defined by giving the value of the gas (vapor) pressure.The atmospheric pressure p0 is equal roughly to 105 Pascals. Pressures below p0 but higherthan about 0.1p0 are termed sometimes negative pressures and they can be measured with theaid of simple mechanical pressure gauges. We meet frequently such ”negative pressures” in ourdaily life, e.g. drinking a cup of liquid. However, in many cases the mean distance between gasatoms is still too small for considering an atom as an isolated one, which can be considered asthe precondition for ”real” vacuum.The ”real” vacuum starts and continues well below 0.1p0. The so-called preliminary vacuumcorresponds to pressures not lower than 10−2 Pa and can be realised with the help of a rota-tional pump (cf. Fig.22-1). Such preliminary vacuum pressures can be measured with thehelp of mechanical pressure gauges, in which the measured gas pressure acts on a mechanicalelement: a fine steel pipe or membrane. Such a gauge records the difference δp between thegas pressure p and the atmospheric pressure p0. Preliminary vacuum corresponds to a situ-ation when the mean free path of gas particles (the mean path between two collisions) is <0.1milimeter, so it’s much smaller than the typical dimensions of the vacuum-setup. This typeo pressure (vacuum) is used in the experiment described below. The rotational pump (of theBL8P type) has a pumping rate of about 2 liters per second and the finale pressure attainedcan be as low as 2 · 10−2 Pa. The basic idea of such a pump is given in Fig.22-1: in a metallicstator there is an eccentric cylindrical rotor, equipped with spring-loaded sliders. The rotationof the rotor alternatively sucks in gas from the to-be-emptied reservoir and pushes it out. Inorder to prevent any unwanted gas-leaks the sliders’ ends and the inner surface of the statorare lubricated with special oil.


Figure 22-1: Rotational pump.

Verification of the Boyle-Mariotte lawThe ideal gas, i.e. a gas whose particles have negligible volume and do not interact, obeys theClapeyron equation:

pV =m

µRT, (22.20)

where: p is the gas pressure, V – gas volume, m – gas mass, T – gas temperature, µ – gasmolecular mass, and R is the universal gas constant, R = 8.313 J/K. For T being constant(T ≡ T0) the above equation reduces to a simpler form:

pV = constant =m

µRT0, (22.21)

known under the name of the Boyle-Martiotte’s law. The vacuum setup (cf. Fig.22-2), con-sisting of several glass chambers of well-defined volume allows to carry out an experimentalverification of this law. By isolating a certain amount of gas in the initial volume (0-th cham-ber) and gradual expansion of this amount of gas into bigger and bigger volumes (consisting ofgreater and greater numbers of chambers involved) one can establish a functional dependencebetween the measured (mechanical gauge) gas pressure and its volume (calculated as a sum ofknown chamber volumes). According to equation 22.21 the dependence should be of the form:p = C/V , with the constant C containing the universal gas constant R.

Setup A. Pumping of the vacuum setup; controlling its the gas-tightness.


1. Examine thoroughly the scheme of the vacuum setup, identify its all components, espe-cially valves.

2. Verify the positions of all valves; at the starting point they all should be closed exceptvalves No. 5 and 6.


Figure 22-2: The vacuum setup.

3. Branch the Pirani’s gauge on.

4. Start the rotational pump; open the No. 3 valve starting simultaneously the measurementof time.

5. Open successively the valves 8 - 17.

6. Observe and record the readings of the mechanical and Pirani’s gauges in the function oftime.Essentially, if the setup is functioning properly (no gas leakage) you should reach a finalpressure of the order of 5 Pa in about 10 minutes.

7. Close the No.3 valve; observe during 10 minutes the variations of pressure (it shouldincrease not more than up to about 50 Pa during that time).

Report

1. Construct a graph p(t) using a semi-logarithmic scale (log p versus t).

B. Verification of the B-M law

Experimental ProcedureThe verification consists in filling up the chamber No.0 with the air under atmospheric pressure(p0) and expanding it gradually into chambers No.1 through No.9. The pressure is measuredafter each expansion.

1. Pump out all the chambers until you reach a pressure of the order of about 50 Pa.

2. Shut off the valves leading to the pump and Pirani’s gauge; close all the valves 8 - 17.

3. Open the valve No.7 (filling the chamber No. 0) and close it again.

4. Open the valve No.8; read out the pressure shown by the mechanical gauge.


5. Start expanding the air by opening the valves 9, 10, . . . , 17; after each expansion readout the value of pressure shown.

6. Read out the atmospheric pressure on the barometer.

7. Pump out the setup, close the valves 9,. . . ,17.

Report

1. Calculate the actual pressures for each expansions (cf. Introduction)

2. Construct a graph p = p(

1

V

). Do not forget to include the error bars for your pi values.

Compare your curve with a ”theoretical” one (a straight line linking two points: the (0,0)point and the (V0, p0) point.)

C. Calculation of the universal gas constant R


1. Establish the mass of air contained in the initial chamber by weighing the chamber (a)empty (with the air pumped-out) and (b) filled with air under atmospheric pressure. Usean analytical balance.

2. Read out the temperature and the p0 value.

Report

1. Calculate the R value knowing T,m and the slope of your p = p(

1

V

)curve. The µ

value should be calculated as a weighted mean; air can be treated as a mixture of gases:N2 (78%), O2 (21%), and Ar (1%). Calculate the error ob R determination, takinginto account the errors on the slope determination and the temperature measurement.Compare the obtained value with the value from physical tables. In case of inconsistencytry to find out possible reasons for the lack of agreement.


• Ideal gas and Clapeyron equation.

• Gas transformations: isothermic, isobaric, isochoric.



Experiment No.25 — INTERFERENCE OF ACOUSTIC WAVES

Purpose:Determination of the sound velocity in gases with the aid of the Quincke’s ”trombone”. Deter-mination of the ratio of specific heats of gases κ = cp/cV .

Introduction:The acoustic waves are waves of the longitudinal type, which can be generated and transportedin any medium: solid, liquid or gaseous. They manifest themselves as in periodic changes ofthe pressure and density of portions of the medium. By convention, as the acoustic waves areconsidered those that belong to the 20 - 20 000 Hz range, as this is the interval of frequenciesaudible to the human ear. However, other ”sounds” (ultra- or infra-sounds), inaudible tohumans, have exactly the same nature as ”normal” acoustic waves.A physical system performing harmonic oscillations generates harmonic (sinusoidal) waves. Thedeviation ∆p of the momentary pressure p(x, t) in any medium at point x and at time t fromthe equilibrium value p0, is given by

∆p = p− p0 ≡ y = ym sin(kx− ωt), (25.22)

where k = 2π/λ and ω = 2π/T are the wave number (wave vector magnitude) and angularfrequency, respectively. As usually, T denotes the wave period and λ – wavelength. Theamplitude of deviations is ym. Actually, equation 25.1 represents a wave traveling in x-direction.

Figure 25-1: Interference scheme for acoustic waves reaching detector via two different routes;S – sound source, D – detector.

Two waves meeting in a point of the medium add up, or interfere. The last term correspondsto the situation when the frequencies of two waves are the same (it implies also the equality ofwavelengths). Two waves coming from the same source and reaching a detector after havingcovered different distances x1 and x2 (cf. Fig.25-1) meet as a sum

y = y1 + y2 = ym1 sin(kx1 − ωt)m + ym2 sin(kx2 − ωt) ≡ A sin(ωt− φ). (25.23)

Both the phase shift φ and the resulting amplitude A can be calculated from simple trigono-metric considerations. Particularly, the amplitude A is given be the expression

|ym1 − ym2| ≤ A =√

y2m1

+ y2m2

+ 2ym1ym2 cos k(x1 − x2) ≤ |ym1 + ym2|. (25.24)


The amplitude reaches then the minimum for

cos k(x1 − x2) = −1, (25.25)

i.e. for x2 − x1 = λ(n− 1

2

), n = 1, 2, . . .. The difference of the distances traveled by the two

waves must be equal to an odd multiple of the half-wavelength: x2−x1 =1

2λ,

3

2λ,

5

2λ, . . .. The

two consecutive ∆x ≡ x1−x2 values differ then by λ. Measuring ∆x (by observing two or moreconsecutive minima of recorded sound intensity) is equivalent to measuring the wavelength and,as the wave frequency ν = 2πω is known, the wave velocity c can be easily calculated as:

c = νλ. (25.26)

On the other hand, the sound velocity in gases is given by the formula

c =

√κRT

µ, (25.27)

with T and µ being gas temperature and molecular mass, respectively; R is the universalgas constant R = 8.313 J/K and κ ≡ cp/cV is the ratio of the molar specific heats for gas atconstant pressure and volume, respectively. Thus, once the sound velocity is known (measured)this parameter can be easily calculated and confronted with its theoretical value (e.g., κ = 7/5for a monoatomic gas, and κ = 7/5 for a diatomic one).

Figure 25-2: A Quincke’s trombone.

Experimental SetupThe experimental setup, known as the Quincke’s trombone, is shown in Fig.25-2 . Waves froman acoustic membrane (activated by a frequency generator) travel through two separate U-shaped pipes. One of the pipes is of a constant length (x1); the length of the other pipe (x2)can be varied (as in a trombone). The detecting system is a pair of headphones; the goal is tofind such values of (x2) that correspond to the maximum damping of the acoustic signal andcalculate the λ value.


Experimental procedure

1. Empty the two pipes by pumping out the gas contained in them and fill them with airby opening the valve.

2. For several (at least 10) acoustic frequencies find 2 - 5 positions of the moveable arm thatcorrespond to the minimum of the acoustic signal; record them as a1, a2, a3, . . . values (Cf.table)

3. Read out the temperature of air.

4. Repeat the whole procedure for the system filled with a different gas; after emptying it(pumping out) you should fill it with the gas supplied by the tutor.

Report

1. Using the table as below record the measured values and calculate the wavelength andwave velocity for each recorded minimum of the sound signal. Calculate the mean (aver-age) value c, and its standard deviation σc. Repeat the calculations of the mean rejectingthe results that differ from the calculated mean c by more than ±2σc. Confront the finalc value with the value given in physical tables. Note: do not forget about the temper-ature dependence of c. The formula 25.27 should be used if the two confronted valuescorrespond to different medium temperatures.

2. Calculate the cp/cV ratio using equation 25.27. The µ value for air should be calculatedas a weighted mean; air can be treated as a mixture of gases: N2 (78%), O2 (21%), andAr (1%).

ν Sound minimum Inter-minimum distance λ c[Hz] [mm] [mm] [m] [m/s]

a1 a2 a3 a4 a5 ∆1 ∆2 ∆3 ∆4


• What is wave? Define quantities: amplitude, phase, period, wavelength, velocity, fre-quency.

• Longitudinal waves and transverse waves.

• Sound and its velocity (if the sound wave propagates in a medium what medium propertiesinfluence the wave speed?

• Interference – general ideas and the particular case (the one you will have to deal withwhen performing the experiment).


Experiment No. 33DIELECTRIC CONSTANT

Purpose:Measurement of the capacitance of capacitors with and without dielectric substance betweenthe plates, in order to determine the permittivity ε0 and dielectric constants κ for variousmaterials.

IntroductionA capacitor is a device for storing energy in an electric field. The basic elements of anycapacitor are two isolated conductors of arbitrary shape — plates. When a capacitor is charged,its plates have equal but opposite charges of +q and −q. The plates being conductors, they areequipotential surfaces: all points on one plate are at the same electric potential. Moreover, thereis a potential difference V between the plates, and the charge q and the potential difference Vare proportional to each other. The capacitor’s capacitance C is the proportionality constantin that relationship:

q = C V (33.28)

The capacitance C is a quantity, whose value depends on the geometry of the plates and onthe nature of the material that fills the inter-plate space. For a parallel plate capacitor, wefind that

C =ε0κA

d, (33.29)

where A is the area of the plate, d is the plate separation, ε0 is the permittivity:

ε0 = 8.85× 10−12F/m (33.30)

and dimensionless κ is the so-called dielectric constant of the material introduced between thecapacitor’s plates (the dielectric constant of a vacuum is unity by definition; that of air ispractically also equal to 1).Formula 33.29 suggests that in order to find the two quantities: ε0 and κ one should measurethe capacitance of capacitors of well-defined (and known) geometrical parameters, empty anddielectric-filled.

Measuring the permittivity constant ε0

The parallel-plate capacitor used in this measurement consists of two circular metal plates.The separation distance d can be adjusted by stacking, in three points, three piles of isolatingdiscs. The value of the capacitance is measured with the aid of a digital capacitance gauge.The formula 33.29 cannot be used directly for determining the permittivity constant ε0 becauseof two reasons. First, the distance-discs are made of a material whose κ has a value muchgreater than unity, so the total capacitance of the system is seriously increased. Therefore, weshould conceive the system as a parallel combination of a capacitor filled with a material whosedielectric constant is κ and total area of the plate equals to 3Ap (where Ap is the area of asingle distance disc) and an empty (air-filled) capacitor, whose total plate area is A−3Ap. Thesystem capacitance is:

C =ε0(A− 3Ap)

d+

3Ap ε0κ

d, (33.31)


and the ε0 can be found as:

ε0 =Cd

A + 3(κ− 1)Ap

. (33.32)

The second, more subtle problem, arises from the so-called edge effects. A simplistic pictureof the electric field in a parallel-plate capacitor is that of a perfectly uniform field (parallellines) inside and no field at all outside. In reality, at the capacitor’s edges the field changesfrom the uniform (inside) to zero (outside) only gradually (cf. Fig.33-1). Around the circularplates (radius r), separated by the d distance, there exists a region whose volume is of the orderof 2πrd2 and which we call the edge or diffused field. This field contributes also to the valueof the capacitor’s capacitance, parallely to the contribution of the uniform filed inside, whosevolume is of the order of πr2d.

Figure 33-1: Electric field in a parallel-plate capacitor.

The relative contribution of the two fields, the diffused field and uniform field, can be takenas a ratio of the two mentioned volumes: 2d/r. Thus, the contribution of the edge field willtend to zero for the separation distance d → 0. This fact suggests an empirical method foreliminating the influence of the edge field: one should perform a series of C measurements withthe d value gradually being decreased and then, construct a plot Cd = f(d), like the one shownin Figure 33-2. The plot points can be than fitted by a continuous curve (a computer simplefitting program can be used) and the value of Cd corresponding to the d = 0 can be found byextrapolation. This extrapolated value should be then inserted in the numerator of 33.32.


Figure 33-2: The method for accounting for the presence of the diffused field (see text).

Measuring the relative permittivity constant κThe relative permittivity constant κ can be determined if one performs two measurements ofthe capacitance of a parallel-plate capacitor: the first for the capacitor being empty, and thesecond for the same capacitor whose plates are separated by a layer of the material whose κ weare looking for. In this case, one should resign from introducing the diffused-field corrections.Another example of a capacitor is a coaxial cable (cf. Fig.33-3). It can be perceived as acylindrical capacitor, whose one ”plate” is the central wire, and the another – the copperbraid. The capacitance of a cylindrical capacitor is given as:

C =2πε0κl

ln Rr

, (33.33)

where R and r are the outer and inner radii, respectively, and l is the capacitor’s length.

Figure 33-3: A coaxial cable.


Measuring the speed of light cMaxwell’s equations, which are considered by many physicists as the most beautiful thingthat ever has been conceived by human mind, contain three physical constants: permittivityconstant, ε0, permeability constant, µ0, and the speed of light c. These three constant, however,are not independent; there exists a relation:

c =1

√ε0µ0

. (33.34)

Moreover, the permeability constant µ0 has an exact by definition value:

µ0 = 4π × 10−7H/m (33.35)

(it has been introduced in order to reconcile the units of electric and magnetic quantities).From equation 33.35 then follows, that measuring ε0 one measures in fact the speed of light,c. The measurements of the speed of light have a long and most interesting history: Galileowas the first scientist who realized that the speed was immense but still a finite one. The firstquantitative measurement is due to Roemer (1676) and the result was about 2/3 of the actualvalue. But already at the beginning of the 18th century c was known with a fair degree ofaccuracy. The first very precise measurements were carried out by Michelson (in the eightiesof the 19th century). Since 1983, however, as part of the redefinition of the meter, the speedof light has been given an assigned value, exact by definition, of

c = 299, 792, 458 m/s.

Experimental SetupTwo metal plates (discs), several isolating discs, and few isolating circular plates. Cylindricalcapacitor (a section of concentric cable). Measuring devices: micrometric screw, yard-stick anda digital capacitance gauge MIC-4070D. The maximal error of the gauge is 1% of the measuredquantity + 0.1% of the range used.


1. Branch on the capacitance gauge.

2. Build up a parallel-plate capacitor, putting one circular disc exactly above the other, andusing three isolating discs. Measure (micrometric screw) the three thicknesses: d1, d2 andd3 of the discs and use an arithmetic mean in the following calculations. Read out thecapacitance of the system.Note: When measuring C keep you hands of the capacitor! By touching it you change(increase) the true capacitance.

3. Repeat the measurement of C for 2,3,4,5,. . . separating discs in each of 3 piles. For eachconfiguration measure the 3 heights of three piles d1, d2 and d3 of the discs and use anarithmetic mean in the following calculations.

4. Measure the capacitance of parallel-plate capacitors built-up as a system of two parallelconducting discs separated by a layer of dielectric.


5. Measure all remaining quantities you need for calculating ε0 (or c) and κ (diameters: Dand Dp).

6. measure the capacitance of the piece of a coaxial cable and its geometrical parameters..

Report

1. Construct the Cd versus d plot. Fit the experimental points with a smooth curve andextrapolate the Cd value for d = 0.

2. Calculate ε0 with the aid of equation 33.32. Find the error of your measurement.Note: you may adopt 2.6 as the value of κ for Plexiglas. Calculate ∆(ε0) for the smallestvalue of d.

3. Calculate the κ values for dielectrics used in your experiment. Confront your results withthe data from physical tables


• Electric field of a static charge.

• Capacity and capacitors – definition, units.

• Capacitors in parallel and in series – formulae.

• Dielectrics – definition and properties. Dielectric constant.

• Velocity of light.


Experiment No.41Mesuring the Earth’s magnetic field (tangent galvanometer)

Purpose:Measurement of the horizontal component of the Earth’s magnetic field with the help of atangent galvanometer.

IntroductionThe Earth magnetic field for points close to the Earth’s surface is essentially a dipole field.The situation is depicted in Fig.41-1. The dipole axis makes an angle of about 11o with theEarth’s rotational axis RR. The poles PG and PM are, respectively, the Earth’s geographicNorth pole and its magnetic North pole. Note, that the latter is actually the South pole ofthe Earth’s magnetic dipole. The magnitude of this magnetic dipole moment is about 8.0×1022

J/T. It is worth to be noted, that because of “practical” reasons (navigation, communication,

Figure 41-1: The Earth’s magnetic field represented as a magnetic dipole field.

prospecting, etc.) the Earth’s magnetic field has been studied extensively for many centuries.However, even today the exact mechanism responsible for the magnetism of the Earth is notquite well understood. The major factor responsible are very intense electric currents flowingin the liquid Earth’s core. The energy necessary for the currents to flow comes – as it seems –from the interaction between Earth and its satellite - Moon. It is worth to be noted that manyitems in the nature and history of the Earth’ magnetism seem still rather mysterious. Such are,for instance, the migration of the magnetic poles and the periodical reversal (with the periodof the order of one million years) of the direction of the field (well confirmed by studies of themagnetic profile of the sea-floor in the vicinity of the Mid-Atlantic Ridge)


The magnitude and direction of the magnetic (vector) field vary depending on the location.The direction can be specified in terms of declination, which is the angle between the “true”geographic north and the horizontal (i.e., tangent to the Earth’s surface) component of thefield) and inclination, which is the angle between the horizontal plane and the field direction.There are a vast number of measuring techniques and tools used to measure theses quantities.The method used in this experiment for measuring the horizontal component of the Earth’smagnetic field is based on the use of the so-called tangent galvanometer, which is a simpledevice consisting of a magnetic needle (magnetic compass) placed in the centre of a verticalcoil. The coil, consisting of several loops of a copper wire is connected to a source of DC.

Magnetic field of electric current

Figure 41-2: (a)A current element i ds establishes a differential magnetic field dB at point P(the field points into the page); (b) circular loop and its magnetic field.

Figure 41-2(a) shows a wire carrying a current i. It is known, that each element of such wireproduces at point P , whose location with respect to ds is given by the vector r, a differentialfield dB. The magnitude and the direction of the dB vector are given by the law of Biot andSavart:

dB =µ0i

4π

ds× r

r3. (41.36)

Here µ0 is a constant, called the permeability constant, whose value is exactly, by definition,4π × 10−7T· m/A.The above formula can be readily used for the case of a circular (radius R) loop of wire (cf.Fig.41-2(b)). The vector ds is, at every point of the loop, perpendicular to the vector r (weare calculating the filed in the very centre of the loop). The absolute magnitude of the fieldcalculated from Eq. 41.36 is (r = R = constant):

dB =µ0i

4π

ds

R2. (41.37)

The integration of this formula is simple – the integral of all the ds elements is simply the loop’scircumference – 2πR. If instead of a single-wire loop we have a coil with n loops the intensity


of the field in the centre of this coil is

B = µ0ni

2R. (41.38)

The direction of the field is, by the virtue of the cross-product, perpendicular to the plane ofthe circle (coil).

The tangent galvanometer, i.e. the loop, the magnetic needle and a goniometric table, can beplaced in such a manner that with no current flowing in the wire the needle should be in theplane of the loop. The needle, of course, is parallel to the Earth’s magnetic field, or rather toits horizontal component B0.By switching on the electric current a magnetic field B is established, with the intensity givenby Eq. 41.38. The two vectors B0 and B are perpendicular; the resulting field Bnet (cf.Fig.41-3) will form an angle α with the field B0. The simple relations are:

B

B0

= tan α; B0 =B

tan α=

ni

2R tan αµ0. (41.39)

Experimental Setup

Figure 41-3: Earth’s magnetic field and the magnetic field produced by coil add up in thetangent galvanometer.

The galvanometer used in the experiment has several connecting points. This allows to usethe galvanometer with different numbers of loops (n = 4, 16, 10) being active. The wholegalvanometer is mounted on a tripod; the regulating screws allow to level horizontally thecompass plane. The remaining parts of the setup are: a slide-resistor, a switch used for reversingthe direction of the current flow and a DC power supply.


Figure 41-4: Experimental setup.


1. Measure the coil radius R.

2. Move the galvanometer as far from the power supply as possible (why?). Use the levellingscrews for horizontal adjustment of the compass plane.

3. Place the coil plane in the plane of the Earth’s magnetic meridian (parallel to the directionof the compass needle).

4. establish the experimental setup (Fig.41-4)

5. Measure the intensity of the electric current for which the needle deflects by angles α(approximately): 30, 40, and 50 degrees.

6. For each angle reverse the direction of the current and measure the mirror angle αm.

7. The results should be entered in the table

No. of loops Current α αm αmean B0

n (A) [o] [o] [o] [Tesla]


Report

1. Calculate the mean value of the horizontal component of the Earth’s magnetic field – B0.

2. Calculate the mean square deviation of all m results

S2 =1

m− 1

m∑i=1

(Bi

0 − B0

)2(41.40)

and the corresponding estimator of the error of the B0 determination.

3. Calculate the error of the determination of a single Bi0 value, taking into account the

errors influencing the determination of the R, I, and α quantities. Compare the twoerrors. Are they consistent?

4. Compare the mean value of the horizontal component of the Earth’s magnetic field – B0

found with the “official value” for Cracow – 21 µT.


• Magnetic induction and magnetic flux.

• Ampere’s Law; Biot-Savart’s Law. The case of a wire-loop (what is the magnetic inductionin its centre?)

• Magnetic dipole and its (magnetic) field. Earth as a magnetic dipole.


Experiment No.51 — INDEX OF REFRACTION FOR SOLIDS

Purpose:Determination of index of refraction for transparent solids (glass, Plexiglas)

IntroductionA beam of light crossing a surface that separates two optically different media becomes bothreflected and refracted at the boundary. In Fig.51-1 an incident light beam is reflected andrefracted by a plane glass surface. The incident beam and the reflected and refracted beamsare represented as rays, which are directed lines drawn perpendicular to the wave fronts andwhich show the direction of movement of the wave. The angle of incidence θ1, the angle ofreflection θ′1, and the angle of refraction θ2 are angles that are measured between the normalto the surface and the appropriate ray. The plane that contains both the incident ray and aline normal to the surface is called the plane of incidence. The laws that govern reflection andrefraction state that the reflected and refracted rays lie also in the plane of the incidence and

θ1 = θ′1 reflectionn1 sin θ1 = n2 sin θ2 refraction (51.41)

where n1 is a dimensionless constant called the index of refraction of medium 1, and n2 is theindex of refraction of medium 2. Equation 51.41 is called Snell’s law.

Figure 51-1: Reflection and refraction at an air-glass interface.

It can be deduced from the Maxwell’s equations, that the index n of refraction of a substance,is equal to c/v, n = c/v, where c is the speed of light in free space (vacuum) and v is its speedin the substance. For all substances but vacuum n depends on the wavelength of the light. Thiseffect is called chromatic dispersion (chromatic refers to a color associated with each wavelength


and dispersion refers to the separation of the wavelengths or colors). The refraction in Fig.51-1does not show chromatic dispersion because the beam is monochromatic, i.e. of a single color(wavelength).For a pair of media we can introduce the concept of the relative index of refraction n21 orsimply n:

n21 ≡ n =n2

n1

=v1

c· c

v2

=v1

v2

. (51.42)

Thus n is simply the ratio of the speed of light in the two media. The ”medium of reference”(medium 1) is usually taken as vacuum or air, as the two media are optically almost identical(the index of refraction for vacuum is unity; that for air is 1.00029). Because of the refractionobjects situated in a medium of a larger optical density appear, when observed from air, tobe closer than in reality. A glass panel seems to be thinner than it really is, objects placedin water seem to be closer to the water surface. Those effects can be explained easily if oneconsiders the way the light beams travel from point O, placed at the bottom of a glass slab (cf.Fig.51-2).

Figure 51-2: Light rays traveling through a glass slab.

The beam OA, incident normally at the boundary, α = 0, suffers no refraction – β = α; thebeam OB forms an angle β with the normal to the boundary in glass, and α (α > β) in air.The two beams upon leaving the glass slab diverge, appearing to originate in the point O1. Thedistance O1A = h constitutes an apparent thickness of the slab, and h < d, where d is the trueslab thickness.In the measurements performed, the light beams are almost perpendicular to the slab surface,i.e. the angles α and β are small. Thus (cf. Fig.51-1):

sin α

sin β≈ tan α

tan β=

AB

hAB

d

=d

h= n. (51.43)


The above equation constitutes the method for determining n. The apparent thickness of theglass slab, h, can me measured as a difference between the two position of the microscope tubefor which one obtains sharp images of lines drawn on the upper and the lower side of the slab.The real slab thickness, d, can be measured with the help of a micrometric screw.

Figure 51-3: A microscope.


1. Mark both sides of the glass plate with a sharply-pointed marker; the marks should betwo perpendicular lines, one placed directly over another.

2. Measure the plate thickness d in the vicinity of the marks using a micrometric screw.

3. Put the plate on the microscope table. Position the microscope tube to get a sharpimage of the lower mark, read out the tube position on the micrometric gauge; repeatthe positioning for the upper mark. The apparent thickness h is the difference of the twogauge’s readings.

4. repeat the measurement of h 6 - 10 times.

5. Repeat the whole procedure (items 1–4) for different materials.

6. For a chosen glass plate carry out the determination of n using a monochromatic light(red and blue filters — 6-10 measurements for each filter).

Report

1. Calculate the index of refraction for each of the above measurements; determine the errorsof the determination of n.

2. Calculate the nred and nblue for the chosen plate and the errors of their determination.


• Reflection and refraction of a wave. Snell’s Law.

• Refraction coefficient.

• Total internal reflection; critical angle.


Experiment No.52INDEX OF REFRACTION FOR LIQUIDS

Purpose:Determination of index of refraction for liquids (sugar solution) with the help of Abbe’s re-fractometer. Measurement of the dependence of n upon the concentration of sugar in thesolution.

IntroductionFigure 52-1 shows rays from a point source S in glass incident on the interface between glassand air. For ray a, which is perpendicular to the interface, part of light reflects at the interfaceand the rest travels through it with no change in direction. For rays b through e, which

Figure 52-1: Total internal reflection of light from a point source for incident angles greaterthan θc.

have progressively larger angles of incidence at the interface, there are also both reflectionand refraction at the interface. As the angle of incidence increases, the angle of refractionalso increases; for ray e it is 90o, which means that the refracted ray points directly along theinterface. The angle of incidence giving this situation is called the critical angle θc. For anglesof incidence larger than θc, such as for rays f and g, there is no refracted ray and all the light isreflected; this effect is called total internal reflection. The Snell’s law for the incident angleequal to θc simplifies to:

sin θc =1

n, (52.44)

where n is the index of refraction for glass.


Experimental — Abbbe’s refractometerThe equation 52.1 may be used for determination of n for liquids from the measurement of thecritical angle θc, for which the total internal reflection occurs.

Figure 52-2: The prism-system in an Abbe’s refractometer.

The heart of an Abbe’s refractometer is a system of two prisms P1 and P2 (cf. Fig.52-2) madeof a glass with a relatively high value of the index of refraction. A layer of the liquid fillsup the free space between the two prisms. (The index of refraction for this liquid must besmaller than that of glass the prisms are made of). The rays coming out of the prism P1 areincident on the interface glass-liquid at different angles and become both reflected and refracted.The refracted rays travel through the prism P2 and leave it after still another refraction. Thedirection of the rays leaving the P2 prism is parallel to the direction of those that enter theP1 one. These rays enter the refractometer eyepiece (L1). However, the rays that are incidentat the P1-liquid interface at angles larger than θc undergo the total internal reflection, i.e.they do not enter the eyepiece. The eyepiece’s field of vision becomes partially darkened. Byturning the system of two prism one should obtain as much contrasted picture of two halvesof the eyepiece as possible – such situation corresponds to the incidence at the critical angle.The refractometer’s calibration makes possible to read out directly the value of the index ofrefraction on an appropriate scale.The above brief description corresponds to the situation when the incident light is a monochro-matic one. In practice we are using a white light, which consists of many wavelengths. Forall substances but vacuum n depends on the wavelength of the light, so to different colorscorrespond (slightly) different θc values. In consequence, the darkness-lightness boundary inthe refractometer’s eyepiece would be rather a blurred streak than a sharp line. This effectof dispersion can be, however, compensated for with the help of special compensation prisms.These prisms may be used for increasing the sharpness of the darkness-lightness division line— actually, the compensation refers to the light wavelength of 589 nanometers (yellow light).


1. Prepare a 10-per-cent sugar solution (ca. 100 g/l).

2. Introduce an adequate portion of this solution into the refractometer’s container; measurethe index of refraction 3-4 times.

3. Repeat the above procedure for reduced solutions (4-5 different solutions reduced by thefactor of 2 with respect to the preceding one)


4. Measure the index of refraction for pure water.

Report

1. Construct a graph: n versus the sugar concentration.

2. Compare the n for pure water with the value that can be found in tables.


• Reflection and refraction of a wave. Snell’s Law.

• Refraction coefficient.

• Total internal reflection; critical angle.


Experiment No.54 GLASS PRISM; CHROMATIC DISPERSION

Purpose:Determination of index of refraction for a glass prism and its dependence on the wavelength ofthe incident light.IntroductionThe Snell’s law of refraction (cf. Experiment No.51)

n1θ1 = n2θ2;n2

n1

≡ n, (54.45)

was developed theoretically by Huyghens. It was his idea that the light traveling in a vacuumwith a certain velocity (actually, c = 3 · 108 m/s) upon entering a transparent medium slowsdown to a velocity v = c/n, where n is the refraction index of the medium. Thus the twoquantities n and v are closely related and they should both depend on the structure of themedium (i.e., pattern of the atomic electrons) and the nature of the incident light (i.e., itswavelength). The relationship

n = n(λ) (54.46)

is called dispersion, or rather chromatic dispersion – cf. Fig.54-1 – , ”chromatic” referring toa color associated with each wavelength of the light and ”dispersion” referring to the separationof the wavelengths or colors. The graph n = n(λ) is referred to as dispersion curve.The classical theory of the dispersion assumes that the light – an electromagnetic wave – inter-acts with atomic electrons, which behave like classical oscillators possessing certain resonancefrequencies ω0. From this theory follows a formula that gives, for a single resonance frequency,the n(λ) or rather n(ω) dependence in the form

n2 − 1

n2 + 2=

constans

ω20 − ω2

. (54.47)

The above formula can be also derived on the basis of a quantum description of the light–material medium interaction. In this case the ω0 value can be associated with the frequenciesat which the incident light becomes strongly absorbed by the medium.It follows from equation 54.3 that the index of refraction increases as the wavelength decreases,since we always have ω < ω0 in practice. Thus, in the visible light range the strongest refraction(the most pronounced ”bending” of the light rays) will be occurring for the violet light.In order to verify equation 54.3 we may substitute for ω:

ω = 2πf = 2πc

λ(54.48)

and take the reciprocals of both members. We obtain

n2 + 2

n2 − 1= constans

(1

λ20

− 1

λ2

). (54.49)

Thus, plotting the ratio (n2 + 1)/(n2 − 1) as a function of 1/λ2 one should obtain a straightline. The point at which the extrapolated line intersects the 1/λ2 axis gives the ω0 at whichthe medium ceases to be transparent to the light.


Figure 54-1: Chromatic dispersion.

Measuring the n(λ) dependence with the aid of a glass prismA ray of light incident on a surface of a glass prism (cf. Fig.54-1) is refracted twice at theair-glass interfaces. If the ray inside the prism is perpendicular to the bisetrix of the prism’svertex angle φ (the so-called symmetric incidence – the emerging ray makes the same angle withthe normal to the prism face as the incident one) the deviation angle α (the angle betweenthe original and final direction of the light beam) has the smallest possible value: α = αminand then it is related to the index of retraction by the formula:

n =sin

αmin + φ2

sinφ

2

. (54.50)

The above formula can be easily deduced from the law of refraction and trigonometrical argu-ments.In order to establish the n = n(λ) dependence measurements of n (using the formula 54.6) arecarried out for light of different colors (wavelengths). A convenient source of monochromaticlight rays of several different wavelengths is a mercury-vapor lamp. The following table liststhe major components of the light emitted by the lamp:


Figure 54-2: A symmetric incidence of light on a glass prism (α = αmin); φ – prism’s vertexangle.

Experimental setupThe experimental setup consists of a mercury-vapor lamp (discharge tube), a glass prism anda goniometer (optical spectrometer). The goniometer major components are: collimator, tele-scope and a rotating table upon which the prism may be placed (cf. Fig.54-3.

Figure 54-3: Block diagram of a prism spectrometer (goniometer).


Table 1: Major light wavelengths occurring in the mercury-vapor lamp’s spectrum.

No Ray color Intensity (position) Wavelength [nm]1 orange weak 6234.42 yellow strong (1st line of a doublet) 579.13 yellow strong (2nd line of a doublet) 577.04 green very strong 546.05 green-blue weak (close to No.6) 496.06 green-blue medium 491.07 violet very strong 435.88 violet weak (close to No.9) 407.89 violet medium 404.7


1. Branch the mercury lamp on. The lamp becomes fully operational (constant light intensityat a highest level) after ca. 5 minutes. During the measurements the lamp should beall the time on. Avoid watching directly the lamp’s slit (the lamp emits a rather strongultraviolet radiation that may be potentially harmful to the eyes).

2. Become familiar with the goniometer, particularly with the angular nonius.

3. Put the prism on the goniometer’s table. Identify the lamp’s lines in accordance withTable 54.1.

4. Find the αmin angle for the green light. With a fixed position of the telescope, rotatethe table and the prism and find the position for which the motion of the green line inthe eyepiece reverses its direction. Leave the prism in this position for all forthcomingmeasurements.

5. Measure the values of αmin for all remaining colors (lines) of the lamp’s spectrum.

Report

1. Calculate (formula 54.4) n for all the lines of the spectrum.

2. Construct a graph: n versus the wavelength.

3. Construct a linearized graph (formula 3) and determine (by extrapolation) the value ofλ0.


• Chromatic dispersion.

• Line spectrum emitted by an (excited) atom.

• Glass prism; how does the light traverse the prism?


Experiment No. 61ELECTROMAGNETIC OSCILLATIONS IN AN RLC CIRCUIT

Purpose:Observations of damped and aperiodic oscillations in an RLC circuit with the help of anoscillator card. Measurements of the parameters that control the observed U(t) dependencies.

IntroductionAn RLC circuit (cf. Fig.61-1) consists of 3 elements: resistance R, capacitance C, and in-ductance L. For a simpler LC circuit it is well known that the charge, current and potentialdifference grow and decay exponentially. In a real circuit, however, a resistance R is alwayspresent (at least as the resistance of the wire the L coil is made of).

Figure 61-1: An RLC circuit.

The differential equation that describes the damped oscillations in an RLC circuit has theform:

Ld2q

dt2+ R

dq

dt+

1

Cq = 0. (61.51)

The three terms correspond to voltage drops at three elements: L, R, and C, respectively, andq is the capacitor’s charge, q = q(t). The function q(t) is measured in the experiment or, moreprecisely, the capacitor’s potential difference U = U(t). These two quantities are, however,linked by a simple U(t) = q(t)/C relation. If we put R = 0, this equation reduces to a simplerform, which describes the undamped LC oscillations, whose angular frequency ω0 is equal to

ω0 =1√LC

. (61.52)

For an RLC circuit equation 61.51 can be readily solved — in fact it is one of the fundamentaldifferential equations of physics. A general solution has the form:

q(t) = A exp(λ1t) + B exp(λ2t), (61.53)


where the two constants A and B depend on the initial conditions. These conditions can beformulated as:

q(t = 0) =U0

C, and

dq

dt

∣∣∣∣∣t=0

= 0. (61.54)

The first condition tells that at the initial moment the capacitor’s charge is equal to the ratioU0/C, where U0 is the voltage of the power supply. The second, states that the current in thecircuit is null (because of the presence of the inductance) at t = 0.The λ1,2 constants in 61.53 are the roots of characteristic equation:

Lλ2 + Rλ +1

C= 0. (61.55)

The character of the solution of equation 61.51 depends on the sign of the discriminant ∆ of61.55, ∆ = R2 − 4L/C ≡ R2 − R2

c , where Rc is the so-called critical resistance and equals

to 2√

L/C. The three possible cases are: ∆ negative, positive and equal to zero, and thecorresponding solutions are discussed briefly below and presented in Figs. 61-2 and 61-3.

R < Rc: damped oscillationsFor a negative ∆ the two roots of equation 61.55 are two complex conjugated numbers:

λ1,2 = − R

2L± i

√1

LC−(

R

2L

)2

≡ −β ± iω. (61.56)

The solution 61.53 takes the form

q(t) = exp(−βt)[A exp(iωt) + B exp(−iωt)]. (61.57)

Transforming to the U(t) dependence, after taking into account the initial condition, one obtains

U(t) = A exp(−βt) cos(ωt + δ). (61.58)

The defined already β and ω :

β =R

2L, ω =

√1

LC−(

R

2L

)2

, (61.59)

are the circuit damping constant and angular frequency, respectively. The two remainingquantities, A and δ can be expressed through the initial values:

δ = − arctanβ

ω, A =

U0

δ. (61.60)

R > Rc: aperiodic oscillationsFor a ∆ > 0 the two roots of 61.55 are real and negative. Substituting β1,2 ≡ −λ1,2 we have:

β1,2 =R

2L±√(

R

2L

)2

− 1

LC, (61.61)


which may be regarded as two different damping constants. The U(t) dependence becomes

U(t) = A′ exp(−β1t) + B′ exp(−β2t), (61.62)

with the two amplitudes A′ and B′ defined as

A′ =−β2U0

β1 − β2

, B′ =β1U0

β1 − β2

. (61.63)

The aperiodic oscillation (i.e., no oscillation at all) is similar to the process of discharging acapacitor in a still simplified version of circuit, namely in an RC circuit. With L becomingnegligibly small (for great values of R) this discharging of the capacitor has the form:

U(t) = U0 exp(−t/τ), (61.64)

where τ = RC is the time constant of the process.

R = Rc: critical aperiodic oscillationsIn this case the two betas are identical: β1 = β2 ≡ β. As it follows from the theory of lineardifferential equations of second order with constant coefficients, in this situation the voltage-time variations are given by:

U(t) = U0(1 + βt) exp(−βt). (61.65)

The most interesting case – damped oscillations – is shown in Fig.61-2. The three solutions areconfronted in Fig.61-3.

Figure 61-2: Damped oscillations: U(t) dependence. The coordinates of the marked points areto be read off from the computer screen during measurements.


Figure 61-3: Confrontation of U(t) dependence for damped (a), aperiodic (b), and (c) criticallyaperiodic oscillations.

Experimental SetupThe RLC circuit (Fig61-1) consists of a capacitor, a coil with a ferrite core, a decade resistor,a 1.2V battery (voltage supply) and a switch. The core of the measuring system is a computerequipped with an oscilloscope card. An extra multimeter makes possible to measure the R, L,and C parameters.


1. Set the RLC circuit as shown in Fig.61-1, but without any extra resistance R. Before,measure the capacitance C of the capacitor and the inductance L of the coil using thedigital multimeter.

2. Branch the computer on, choose the recording time of the order of one second.

3. Charge the capacitor, release the computer recording of the voltage, and move the switchto the ”off” position immediately.

4. Measure the coordinates of the points shown in Fig.61-2 for the damped oscillations,which one observes without any extra resistor (the only resistance R is that of the coilwire).

5. Insert the decade resistor into the circuit (Fig.61-1).

• Calculate Rc = 2√

L/C, set the decade resistor in such manner that the totalresistance of the circuit R is equal to Rc.

• Investigate the critical aperiodic oscillations (for R = Rc).


• Increase R; investigate the aperiodic oscillations (for R > Rc).

• Investigate the damped oscillations with an increased value of R (R < Rc).

• Observe the discharging of the capacitor with the coil removed from the circuit.

Report

1. For the damped oscillations: calculate the apparent period T and angular frequency ω.

2. For the damped oscillations: construct a graph lg |Ui| = f(ti) for the extreme values ofthe voltage.

3. For aperiodic case: construct a graph lg U = f(t).

4. Identify the linear section of the graphs; calculate the slope and the value of the dampingcoefficient (β or β2

1). For the damped oscillations make use of the least squares methodfor fitting a straight line to the graph | lg Ui| = f(ti); for the aperiodic oscillations fit astraight line using manual method.

5. For the critical aperiodic case superimpose on the experimental graph a theoretical graphcalculated with the aid of equation 61.62.

6. Confront the measured (from the graphs) values of ω, β, and β2 with the values calculatedfrom the known (measured with the help of multimeter) R, L, and C values.


• RLC circuit and its differential equation.

• Kirchoff’s Laws.

1β1 > β2, so it is the smaller damping coefficient that governs the time dependence U(t) for larger t values.


Experiment No.71DIFFRACTION AND INTERFERENCE OF LASER LIGHT

Purpose:Measurement of the intensity of laser light in a diffraction-interference pattern originated by asingle-and double-slit.

IntroductionDiffraction and interference are the effects that can be only explained on the basis of the wavenature of light. Both are superposition effects, in that they result from the combination ofwaves with differing phases at a given point. If the combining waves originate from a finite (andusually small) number of elementary coherent sources we call the process interference; if thecombining waves originate in a single wave front that we have divided into coherent sources ofdifferential size we call the process diffraction. Very often these two effects both are presentsimultaneously (e.g., in an experiment with two slits having finite widths).

Figure 71-1: Interference of light on a system of two vanishingly narrow slits.

Double-slit experiment (vanishingly narrow slits)An idealized scheme of the famous Young’s experiment is shown in Fig.71-1. Two slits, each ofa width equal to a, and separated by a distance b are at a distance l from a screen. The distancel is large if compared with b (l � b); in turn, the separation distance b is also much larger than


the silts’ width a (b � a). We are interested in the intensity of a point P of the screen; theposition of P can be expressed either in terms of the angle Θ or the off-center displacement x.To find the intensity we need the phase relationships between the wavelets arriving at P fromboth slits. The path length difference is

PS1 − PS2 ≈ S2D = b sin Θ, (71.66)

so the phase difference ∆φ is given by

∆φ =2π

λb sin Θ, (71.67)

and the electromagnetic wave at P can be expressed as a sum of two contributions:

E = E0 sin ωt + E0 sin(ωt + ∆φ), (71.68)

where the E is the amplitude of two waves and ω is their angular frequency. By simple complexalgebra one can show that the resulting wave is:

E = 2E0 cos

(∆φ

2

)sin(ωt + β). (71.69)

The ”new” phase shift β can be easily expressed by the parameters of the two waves and ∆φ.However, as we are concerned about the intensity at P we need only the amplitude of thecombined wave, as the intensity is directly proportional to the square of the wave’s amplitude.From 71.69 we have

I ∝ cos2

(∆φ

2

). (71.70)

Taking into account the fact that normally x � l we have sin Θ ≈ x/l; introducing this resultinto 71.67 we may write down the final formula for the light intensity at x in the form:

I(x) = I0 cos2

(πbx

lλ

). (71.71)

For two very narrow slits the interference pattern on the screen will be a system of alternatingbright and dark fringes (cf. Fig.71-2). The central points xmax of the light fringes must fulfillthe relation

xmax =mλl

d, (71.72)

where m is an integer; m = 0, 1, 2, . . .. (The cosine attains its maximal value for its argumentequal to a multiple of π.)

Single-slit diffractionThe expression for the intensity I of the diffraction pattern generated by a light illuminatinga single slit can be found in a similar way. One has to divide the slit width a into, say, Nseparate zones of equal widths ∆x that are so small we can assume each zone acts as a sourceof Huyghens’ wavelets, and then superimpose all N wavelets arriving at P with different phases.The appropriate formula (see Halliday, Resnick, Walker for details) is

I(x) = I0

(sin α

α

)2

, with α =1

2∆φ =

πa

λsin Θ ≈ πax

λl. (71.73)


Figure 71-2: Interference or/and diffraction patterns. (a) two very narrow slits; (b) single slit;(c) two slits with the a/b ratio equal to 0.3.

A plot of equation 71.73 is given in Fig.71-2b. The pattern is relatively a simple one: a central,well pronounced, bright fringe with much less pronounced lateral bright fringes distributedsymmetrically. Between the bright fringes, dark ones occur at the points xmin that correspondto minima of 71.73, i.e. for the points (α equal to a multiple of π):

xmin = ±mlλ

a, m = 1, 2, . . . . (71.74)

The integer m can be regarded as a number of the dark fringe. Consequently, the lateralmaxima (α equal to an odd multiple of π/2) will occur at the points:

xmax = ±(m +

1

2

)lλ

a, m = 1, 2, . . . . (71.75)

From 71.73 follows that the observed intensities at the locations of bright fringes are:

I(xmax) = I01

π2(m + 1

2

) . (71.76)


As a measure of the fringe’s intensity spread one can introduce also the so-called full width athalf maximum (FWHM) — a distance between two points situated to the left and to the rightof maximum, at which the fringe intensity amounts to 50% of that of the intensity at the verycenter. Inserting into 71.73 I(x) = I0/2 and solving for x one obtains

FWHM = 1.437lλ

a. (71.77)

Double-slit experiment (non-vanishingly narrow slits)In an actual experiment the slit width a is comparable to the slit separation b (Fig.71-2c). Theintensity pattern results both from interference and diffraction. The appropriate formula, as itcan be shown, is a product of two former ones:

I(x) = I0

(sin α

α

)2

cos2 β, (71.78)

with α and β defined as previously:

α ≈ πax

lλ, and β ≈ πbx

lλ. (71.79)

Note the symmetry of the above definitions; the two factors in 71.78 correspond to diffraction(a – slit width) and interference (b – slit separation) effects. The interference factor cos2 β isdue to the interference between two slits with slit separation b. The diffraction factor [sin α/α]2

is due to the diffraction by a single slit of width a. For two slits of finite width the uniformpattern of Fig.71-2a becomes modulated by the curve of Fig.71-2b that acts as an envelope(Fig.71-2c).It may be noticed that equation 71.13 is a generalization of the previous results and for α → 0,when α → 0 and consequently sin α/α → 1, we recover equation 71.6. For b → 0 equation71.13 becomes 71.8.

Experimental SetupThe source of light used in the experiment is a He-Ne or semiconductor laser, which suppliesmonochromatic, coherent, and parallel light. The light detector is a photodiode, which underthe action of incident light generates an electrical current proportional to light’s intensity. Thegenerated current produces a voltage drop on the resistor R, which can be measured with thehelp of a digital voltmeter (cf. Fig.71-3). Note, that the active area of the photodiode is asquare whose side is approximately 0.8mm. This means, that the light detector’s readingsrepresent an information that has become averaged on the interval ∆x approximately 0.8 mmwide. This explains why the voltmeter readings are never null and the intensities at maximaare slightly reduced with respect to the theoretical values, corresponding to formula 71.78. Thiseffect is illustrated in Fig.71-4.

A. Double slit experiment


1. Get familiar with the experimental setup; connect the light-detector circuit (do notbranch the power supply unit on!)


Figure 71-3: Diffraction and interference – a measuring setup.

Figure 71-4: Single slit intensity measurement. Dashed line – theory (eq. 71.78); solid line –experiment (effect of finite dimensions of the light detector).

2. In presence of a tutor branch the power supply and laser on.

3. Choose a double slit; arrange the laser-slit configuration so the light illuminates the verycenter of the slit.

4. Verify the alignment of your laser-slit-screen geometry. Observe the fringe pattern on the


screen. It should be symmetrical with respect to the central (i.e., the brightest) point.

5. Carry out the light intensity measurements for several fringes (over and below the centralfringe). Use a 0.2 millimeter measurement step.

6. Branch laser and power supply off; measure the slit-screen distance. Write down the valueof the wavelength of emitted light.

Report

1. Construct a graph: U = U(x), where U is the recorded voltage and x is the detectorposition (as measured with respect to the central point). Connect the measurementpoints by a smooth curve, superimpose the diffraction envelope. Read of the locations oftwo maxima; calculate the slit separation.

B. Single slit experiment


1. Insert a variable-width slit into the experimental setup. Align the position of the laser sothat the central point of the slit is illuminated. Adjust the slit width in such a mannerthat few bright lateral fringes are visible.

2. Carry out the U = U(x) measurement as for the double-slit experiment.

Report

1. Construct a graph: U = U(x), where U is the recorded voltage and x is the detectorposition (as measured with respect to the central point). Use a semi-logarithmic scale.

2. Read out from the graph:(a) the FWHM value;(b) locations of the first (left and right) minimum;(c) locations of the first (left and right) maximum;(d) as (b) and (c) – for the second extremum.

3. Calculate the slit width a from (all) above data. Calculate the mean value a and its error,σ(a).

4. Calculate the Imax/I0 ratios from the measurement data. Compare your results with thepredictions of 71.78


• Diffraction (single slit).

• Describe how can we determine the slit width from the experiment data.

• Two slits; Young’s experiment.


Experiment No. 92ABSORPTION OF GAMMA RAYS

Purpose: Measurement of the intensity of a gamma-ray beam passing through different materialsof varying thickness; establishing the intensity-versus-thickness relationship for these materials;determining their gamma-ray absorption coefficients and energy of the gamma-ray source.

IntroductionThe so-called gamma radiation is electromagnetic radiation, arising during transitions of excitednuclear states into states of lower energy (mainly the ground ones). This radiation accompa-nies usually all alpha and beta nuclear decays and other nuclear transmutations, e.g. nuclearreactions. It constitutes the basic de-excitation mechanism for nuclei that happen to be in anexcited state. The gamma photons for typical nuclear inter-state transitions are of the order ofa 1 megaelectronovolt.A source of gamma rays can be, for instance, a cesium isotope – 137Cs. This isotope undergoesthe so-called two-channel beta decay, transforming into barium-137 isotope. However, only8% of decaying cesium nuclei transform directly into barium nuclei in ground state; in theremaining 92% of decays the resulting barium nuclei are in an excited state and they loose theexcess energy by emitting a gamma photon of energy equal to 0.661 megaelectronovolts. Thedecay scheme of cesium-137 is presented in Fig.92-1.

Figure 92-1: Decay scheme of cesium-137 nucleus.

Because of their rather high energy, the gamma rays have high ”penetrating” qualities. How-ever, a beam of gamma photons traveling through a medium always interacts with it throughthree major different processes: photoelectric absorption (ph), Compton scattering (C), andpair creation (pair). These effects lead to removal of a gamma photon from the incident beam(ph, pair) or to degradation of its energy (C).


Photoelectric absorption is a well-known process, in which a photon interacting with anatom removes an electron from one of the inner electron shells (the most probable candidateis always a K-shell electron). The photon vanishes; the kinetic energy of the electron emittedequals to the energy hω of incident photon minus the binding energy Be of the electron at agiven shell:

hω =

(mv2

2

)electron

+ Be. (92.80)

The above simple equation (known sometimes as Einstein equation) gives uniquely energeticdependencies. A simple analysis, based on the momentum conservation principle, shows thatthe phenomenon cannot occur for free electrons – only for electrons bound in an atom. Thus,the phenomenon is more likely to occur for well-bound electrons, i.e. the ones that are close tonuclei of higher Z, and not–too–energetic photons. Therefore, the probability of photoelectricabsorption depends strongly on atomic number of medium nuclei (it is roughly proportional toZ4.5), and photon energy (inversely proportional to hω3.5).Compton scattering on the other hand is a process in which electrons that are being en-countered on the path of the gamma beam are free, or – practically – very weakly bound.The phenomenon can be described formally as a ”normal” elastic collision of a gamma particle(gamma photon) of energy hω and momentum hω/c with an electron at rest. One must remem-ber, however, that the description must be carried out taking into account relativistic effects,namely the relativistic mass increase of the electron. The probability of Compton scatteringis inversely proportional to the photon energy and it depends on (is proportional to) the totalnumber of electrons ne in a unit volume of the interacting medium

ne = ρZ

ANA, (92.81)

where Z and A are atomic and mass number of the medium, respectively; ρ is the mediumdensity and NA – Avogadro’s number. From the above formula follows that the probabilitydepends rather insignificantly on the elemental nature of the medium (Z/A ≈ 0.5 for mostelements).Pair creation is a process that can be only explained on the basis of quantum mechanics:a gamma photon, traveling in a vicinity of a heavy nucleus, disappears and a pair electron-positron is created. From the energy conservation principle follows that the effect is the so-calledthreshold effect, i.e. it occurs only for gamma energies higher then 2 ×mec

2, where me is theelectron (positron) mass and c – the velocity of light. The excess of energy of incident photonis distributed, more or less evenly, among two created particles as kinetic energy.


Figure 92-2: Mass absorption coefficients, µmass, for Al, Pb, and Cu versus gamma- ray energy(solid lines); dashed lines – partial coefficients for Pb.

The three phenomena – ph, C, pair –contribute, with different weights, to the total interactionprocess. The quantity that describes quantitatively the interaction (its probability) is the so-called total absorption coefficient µ and from the above discussion follows that it is a sum ofthree terms

µ = µph + µC + µpair, (92.82)

where the three partial coefficients: µph, µC , and µpair represent the contributions of threeelementary processes described above. This quantity, the total absorption coefficient, may beused for formulating the so-called absorption law for gamma rays:

I(x) = I0e−µx. (92.83)

Here, I0 denotes the intensity (number of photons per unit time) of photons incident on a slab ofmaterial of thickness x, and I is the intensity of the photons that have passed through the slab.It should be, however, emphasized, that the above law applies only for very well collimatedincident and passed-through beams (the so-called ”pencil-beam” geometry), for which anyCompton-scattered photon is irrevocably ”removed” (absorbed) from the incident beam. Inpractical situations, equation 92.83 can be treated only as an approximation.It follows from 92.83 that the total mass absorption coefficient is a quantity whose unit is m−1,or cm−1. For practical reasons, it is convenient to introduce the so-called mass absorptioncoefficient

µmass =µ

ρ(92.84)


a quantity that for the Compton-dominated range of gamma-ray energies (0.5 - 2 MeV) dependsrather weakly on the elemental nature of medium.The graphs of the mass absorption coefficients, µmass, versus gamma-ray energy are shown inFig.92-2

It is clear, that once the (linear) absorption coefficient µ has been measured (using equation92.83), one may calculate the mass absorption coefficient µmass from equation 92.84 and usethe graphs in Fig.92-2 for reading out the source energy.

Experimental SetupThe experimental set scheme is presented in Fig.92-3. It consists of a radiation counter of theGeiger-Mueller type, powered by a high-voltage supply, and connected through a pre-amplifierto a pulse discriminator and scaler. The gamma-ray source is placed in a special housing (leadcontainer), which serves also as a collimator of the incident beam. The different thickness ofabsorbing media is realized with the help of sets of appropriate plates made of lead, aluminumand copper. The individual plate thickness is measured with the aid of a micrometric screw.

Figure 92-3: Experimental setup for measuring the gamma-ray absorption.

Geiger Mueller counterGeiger-Mueller counter belongs to the class of the so-called gas detectors of nuclear radiation.A G-M counter consists of a cathode, which is a cylinder made of a conducting material, andanode, which is a fine wire of a diameter of about 0.01 mm, placed along the cylinder axis.The volume of a gas detector is filled with mixture of different gases, depending on the kind ofmeasurements to be performed. Geiger-Mueller counters for gamma-ray detection usually arefilled with mixtures noble gases (argon) and organic vapors (i.e., methane).


The operating voltage for a typical G-M counter is equal to about 1000 V. An incident gamma-ray knocks out a photoelectron from the cathode wall (in order to increase the probabilityof the photo-emission the cathode wall can be covered with a Bi or W foil). The ejectedelectron, moving under the action of the counter’s electric field, ionizes the noble gas atoms.The electrons ejected during the ionization acts become accelerated on their turn. Also, thegas atoms become excited in the strong electric field and emit X-rays, which again may takepart in subsequent acts of photoelectric absorption. As a results, an electron avalanche, orrather a cascade of electron avalanches builds up, and the bulk negative charge reaches theanode causing a sharp voltage drop at an appropriately connected resistor. Such voltage dropis recorded as a voltage pulse, testifying to the presence of incident gamma-rays. After havingproduced a single pulse, the so-called quenching mechanism starts up (the organic vapors areinvolved here) and after a time of the order of hundred of microseconds the counter returns tothe operational status, i.e. it is ready to record a new pulse.Normally, a G-M counter starts being operational at a HV of about 1000-1200 V, and its effi-ciency (number of pulses recorded under the influence of a radiation beam of constant intensity)remains practically constant for voltages higher by 200-300 V (the so-called ”voltage-plateau”).Before starting measurements with a G-M counter one should determine the starting voltageand the range of the plateau. The measurements should be carried out with the voltage set ina middle of the plateau.


1. Place the source container directly under the G-M counter. The distance source (colli-mator) outlet-counter should be about 10cm. Remove the container lid.

2. Set the HV supply switch at zero volts. Branch all the segments of the set-up on. Startthe scaler. Slowly increase the high voltage (do not overcome 1400 V !), until youobserve the scaler to start counting the pulses. Lower the high voltage and find thestarting point (corresponding voltage) once again.

3. Perform about 10 measurements of the radiation intensity, beginning with the startingvoltage and increasing each time the operating voltage for 20 V. The measurement timeshould be of the order of 30 seconds. If the registered intensity increases sharply for asubsequent measurement do not continue.

4. Set the operating voltage about 100 V above the starting point. Carry out the measure-ment of the background, i.e. the intensity of radiation with the source well shut andplaced at 1 meter distance from the counter (at least). This measurement should be donewith the measuring time of the order of 500 seconds.

5. Bring the source back and place it under the counter. Open the lid. Carry out theabsorption measurements for at least two sets of plates out of three available (Pb, Al, andCu). The measurements should be carried out with the measuring time selected in sucha manner that the total count is not lower than 1000. Start with measurement withoutany absorbing plate and carry on placing on the source container subsequent plates. Thethickness of each plate should be measured with the help of micrometric screw. Theresults should be gathered in the table


Absorber Thickness Total Number Time Intensity Intensity(No. of plate) of plate thickness of counts [s] [cps] - background

[cm] [cm] [cps ]

Report

1. For the established intensity-versus-thickness measurements construct the appropriategraphs. First, linearize the observed relationships; from eq. 92.83 it is easy to see, that

ln I(x) = ln I0 − µx. (92.85)

Do not forget about the measurement errors in the case of radiation intensities.

Hint. The measured intensities I are the ratios: I = N/t, where N is the number ofcounts registered during time t. Assume that only N is subject to a statistical error:

σ(N) =√

N . So σ(I) = σ(N)t

=√

Nt

.

For linearized relationships find the parameters of the straight line(s). The slope(s) willbe the linear absorption coefficient(s), µ.

2. Calculate the mass absorption coefficients, µmass. Find the energy of the gamma-raysfrom the graphs in Fig.92-2.


• Radioactive decays (transformations): alpha decay and beta decay.

• The decay equation. What is the half-life time?

• Gamma radiation as the radiation coming from excited nuclei. What is its nature andcharacteristic energies?

• Radiation absorption law and absorption coefficient.


Physics Laboratory - Wydział Fizyki i Informatyki Stosowanejlenda/ist_phys.pdf · Experiment No.11...

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