From last time… Voltage drop across inductor · 2009-11-19 · 1 Tue. Nov. 9, 2009 Physics 208,...

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Tue. Nov. 9, 2009 Physics 208, Lecture 20 1

From last time…

Inductors in circuits

Inductors Flux = (Inductance) X (Current)

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Φ = LI

Vbatt R

L

I Vb

Va

Voltage drop across inductor   Constant current

  No voltage difference   Current changing in time

  Voltage difference across inductor


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ΔVL =Vb −Va = −L dIdt

RL Circuit

What is voltage across L just after switch closed?


  Before switch closed, IL = 0   Current through inductor cannot ‘jump’   Just after switch closed, IL= 0.

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VL = −L dIdt

A.  VL = 0

B.  VL= Vbattery

C.  VL= Vbattery / R

D.  VL= Vbattery / L

Kirchoff’s loop law:

VR + VL = Vbattery

R and L in series, IL=0 IR=0, VR=0


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VL t = 0( ) = −Vbattery = −L dILdt

⇒dILdt

=Vbattery

L

IL

IL(t)

Time ( t ) 0

0

Slope dI / dt = Vbattery / L

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IL t = 0( ) = 0

IL instantaneously zero, but increasing in time

Just a little later…

A short time later ( t=0+Δt ), the current is increasing …


IL(t)

Time ( t ) 0

0

Slope dI / dt = Vbattery / L

A.  More slowly

B.  More quickly

C.  At the same rate

IL>0, and IR=IL

VR≠0, so VL smaller

VL= -LdI/dt, so dI/dt smaller

Switch closed at t=0


IL

IL(t)

Time ( t ) 0

0

Initial slope

What is current through inductor in equilibrium, a long time after switch is closed?

Later slope

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dIdt

=Vbattery

L€

dIdt

=Vbattery

L− IL t = Δt( )R

A.  Zero

B.  Vbattery / L

C.  Vbattery / R

Equilibrium: currents not changing

dIL / dt =0, so VL=0

VR=Vbattery

IL = IR =Vbattery / R

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Vbattery

R

2


RL summary

I(t)

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I t( ) = I∞ 1− e− t /(L /R )( ) = I∞ 1− e

−t /τ( )

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I∞ =Vbattery /Rτ = L /R = time constant

I(t)

Switch closed at t=0

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I∞ =Vbattery /R

Question What is the current through

R1 immediately after the switch is closed?


L R1

R2

A.  Vbattery / L

B.  Vbattery / R1

C.  Vbattery / R2

D.  Vbattery / (R1+R2)

E.  0

IL cannot ‘jump’. IL=0 just after closing switch.

All current flows through resistors.

Resistor current can jump.

Thinking about electromagnetism

  Many similarities between electricity, magnetism   Some symmetries, particularly in time-dependence


Electric Fields

Arise from charges Capacitor, Q=CV

Arise from time-varying B-field Inductor, Faraday effect

Magnetic Fields

Arise from currents Inductor, Φ=LI

Arise from time-varying E-field


Maxwell’s unification   Intimate connection

between electricity and magnetism

  Time-varying magnetic field induces an electric field (Faraday’s Law)

  Time-varying electric field generates a magnetic field

This is the basis of Maxwell’s unification of electricity and magnetism into Electromagnetism

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∇ × E = − 1

c∂ B ∂t

∇ × B = 1

c∂ E ∂t

In vacuum:


• A Transverse wave.

• Electric/magnetic fields perpendicular to propagation direction

• Can travel in empty space

f = v/λ, v = c = 3 x 108 m/s (186,000 miles/second)


The EM Spectrum

  Types are distinguished by frequency or wavelength

  Visible light is a small portion of the spectrum

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Sizes of EM waves   Visible light

  typical wavelength of 500 nm = = 0.5 x 10-6 m = 0.5 microns (µm)

AM 1310, your badger radio network, has a vibration frequency of 1310 KHz = 1.31x106 Hz

What is its wavelength?

A.  230 m

B.  0.044 m

C. 2.3 m

D. 44m Tue. Nov. 9, 2009 Physics 208, Lecture 20 14

A microwave oven irradiates food with electromagnetic radiation that has a frequency of about 1010 Hz. The wavelengths of these microwaves are on the order of

A. kilometers

B. meters

C. centimeters

D. micrometers

Quick Quiz

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" = c / f =3#10

8m /s

1010/s

= 3cm


Mathematical description

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Bo = Eo /c

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E = E o cos kz −ωt( )

B = B o cos kz −ωt( )

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E ⊥ B

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k =2πλ

, ω = 2πf

z

x

y

Propagation direction =

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E × B


EM Waves from an Antenna

  Two rods are connected to an ac source, charges oscillate between the rods (a)

  As oscillations continue, the rods become less charged, the field near the charges decreases and the field produced at t = 0 moves away from the rod (b)

  The charges and field reverse (c)   The oscillations continue (d)


Detecting EM waves

FM antenna AM antenna

Oriented vertically for radio waves


Transatlantic signals

Gulgielmo Marconi’s transatlantic transmitter

Capacitor banks Induction

coils Spark gap

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Transatlantic receiver Left to right: Kemp, Marconi, and Paget pose in front of a kite that

was used to keep aloft the receiving aerial wire used in the transatlantic radio experiment.


Energy and EM Waves Energy density in E-field Energy density in B-field

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uE = εoE2 r,t( ) /2

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uB = B2 r,t( ) /2µo

Total

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uTot = εoE2 /2 + B2 /2µo

= εoE2 /2 + E 2 /2c 2µo = εoE

2 r,t( ) = B2 r,t( ) /µo

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uTot = εoE2 = εoEo

2 cos2 kz −ωt( ) moves w/ EM wave at speed c


Power and intensity in EM waves   Energy density uE moves at c

  Instantaneous energy flow = energy per second passing plane   =   This is power density W/m2

  Oscillates in time   Time average of this is Intensity =

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cεoEmax2 /2 = cBmax

2 /2µo€

cuTot = cεoE2 = cεoEo

2 cos2 ωt( )


Example: E-field in laser pointer

  3 mW laser pointer.

  Beam diameter at board ~ 2mm

  Intensity =

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10−3Wπ 0.001m( )2

= 318W /m2

  How big is max E-field?

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cεoEmax2 /2 = 318W /m2

Emax =2 318W /m2( )

3×108m /s( ) 8.85 ×10−12C2 /N ⋅m2( )= 489N /C = 489V /m


Spherical waves   Sources often radiate EM wave in all directions

  Light bulb   The sun   Radio/tv transmission tower

  Spherical wave, looks like plane wave far away   Intensity decreases with distance

  Power spread over larger area

 

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I =Psource4π r2

Source power

Spread over this surface area


Question A radio station transmits 50kW of power from its

antanna. What is the amplitude of the electric field at your radio, 1km away.

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I =50,000W4π 1000m( )2

= 4 ×10−3W /m2

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cεoEmax2 /2 = 4 ×10−3W /m2

Emax =2 4 ×10−3W /m2( )

3×108m /s( ) 8.85 ×10−12C2 /N ⋅m2( )=1.73N /C =1.73V /m

A.  0.1 V/m

B.  0.5 V/m

C. 1 V/m

D. 1.7 V/m

E.  15 V/m

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The Poynting Vector   Rate at which energy flows through a unit area perpendicular

to direction of wave propagation

  Instantaneous power per unit area (J/s.m2 = W/m2) is also

  Its direction is the direction of propagation of the EM wave

  This is time dependent   Its magnitude varies in time   Its magnitude reaches a maximum at the

same instant as E and B

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S = 1

µo

E × B ≡ Poynting Vector


Radiation Pressure   Saw EM waves carry energy   They also have momentum   When object absorbs energy U from EM wave:

  Momentum Δp is transferred

 

  Result is a force

  Pressure = Force/Area = €

Δp =U /c ( Will see this later in QM )

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F = Δp /Δt =U /Δtc

= P /c

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prad =P /Ac

= I /c

Radiation pressure on perfectly absorbing object

Power

Intensity


Radiation pressure & force EM wave incident on surface exerts a radiation pressure prad (force/area) proportional to intensity I.

Perfectly absorbing (black) surface:

Perfectly reflecting (mirror) surface:

Resulting force = (radiation pressure) x (area) €

prad = I /c

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prad = 2I /c


Question A perfectly reflecting square solar sail is 107m X 107m. It has

a mass of 100kg. It starts from rest near the Earth’s orbit, where the sun’s EM radiation has an intensity of 1300 W/m2.

How fast is it moving after 1 hour?

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prad = 2I /c

Frad = prad A = 2IA /c =2 1300W /m2( ) 1.145 ×104m2( )

3×108m /s= 0.1N

a = Frad /m =10−3m /s2

v = at = 10−3m /s2( ) 3600s( ) = 3.6m /s

A.  100 m/s B.  56 m/s C. 17 m/s D. 3.6 m/s E.  0.7 m/s

From last time… Voltage drop across inductor · 2009-11-19 · 1 Tue. Nov. 9, 2009 Physics 208,...

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Transcript of From last time… Voltage drop across inductor · 2009-11-19 · 1 Tue. Nov. 9, 2009 Physics 208,...