Prefixes

a=10−18, f=10−15, p=10−12, n=10−9, µ = 10−6, m=10−3, c=10−2, k=103, M=106, G=109, T=1012, P=1015

Physical Constants

k = 1/4πǫ0 = 8.988 GN·m2/C2 (Coulomb’s Law) ǫ0 = 1/4πk = 8.854 pF/m (permittivity of space)

e = 1.602× 10−19 C (proton charge)

me = 9.11× 10−31 kg (electron mass) mp = 1.67× 10−27 kg (proton mass)

Units and Conversions

NA = 6.02× 1023/mole (Avogadro’s #) 1 u = 1 g/NA = 1.6605× 10−27 kg (mass unit)

1.0 eV = 1.602× 10−19 J (electron-volt) 1 V = 1 J/C = 1 volt = 1 joule/coulomb

1 F = 1 C/V = 1 farad = 1 C2/J

1 A = 1 C/s = 1 ampere = 1 coulomb/second 1 Ω = 1 V/A = 1 ohm = 1 J·s/C2

Vectors

Written ~V or V, described by magnitude=V , direction=θ or by components (Vx, Vy).

Vx = V cos θ, Vy = V sin θ,

V =√

V 2x + V 2

y , tan θ =Vy

Vx

. θ is the angle from ~V to +x-axis.

Addition: A+B, head to tail. Subtraction: A−B is A+ (−B), −B is B reversed.

Trig summary

sin θ = (opp)(hyp) , cos θ = (adj)

(hyp) , tan θ = (opp)(adj) , (opp)2 + (adj)2 = (hyp)2.

sin θ = sin(180 − θ), cos θ = cos(−θ), tan θ = tan(180 + θ), sin2 θ + cos2 θ = 1.

Chapter 16 Equations

Charges:

Q = ±Ne, ∆Q1 +∆Q2 = 0, e = 1.602× 10−19 C.

Electric Force:

F = kQ1Q2

r2 , k = 8.988× 109 N·m2/C2, F = Q1Q2

4πǫ0r2, ǫ0 = 1

4πk = 8.854 pF/m.

~F = ~F1 + ~F2 + ~F3 + ... superposition of many forces.

Fx = F1x + F2x + F3x + ... superposition of x-components of many forces.

Fy = F1y + F2y + F3y + ... superposition of y-components of many forces.

Electric Field:

~E =~Fq , q= test charge. Or: ~F = q ~E.

| ~E| = E = k Qr2 = Q

4πǫ0r2, due to point charge. Negative Q makes inward ~E, positive Q makes outward ~E.

~E = ~E1 + ~E2 + ~E3 + ... superposition of many electric fields.

Ex = E1x + E2x + E3x + ... superposition of x-components of many electric fields.

Ey = E1y + E2y + E3y + ... superposition of y-components of many electric fields.

E = k Qr2 = electric field around a point charge or outside a spherical charge distribution.

Eq.-1

Chapter 17 Equations

Potential Energy and Work:

Wba = FEd cos θ = work done by electric force FE on test charge, in displacement d from a to b.

Wba = −q∆V = −q(Vb − Va) = work done by electric force on a test charge, moved from a to b.

∆PE = q∆V = q(Vb − Va) = change in electric potential energy of the system. Also: ∆PE = −Wba.

∆KE +∆PE = 0, or, ∆KE = −∆PE = −q∆V , principle of conservation of mechanical energy.

Potential:

∆V = ∆PEq = definition of change in electric potential.

∆V = Ed = potential change in a uniform electric field.

V = kQr = potential produced by a point charge or outside a spherical charge distribution.

PE = qV = potential energy for a test charge at a point in a field.

PE = kQ1Q2

r12= potential energy of a pair of charges.

Capacitance:

Q = CV , C = Kǫ0Ad = capacitor equations.

U = 12QV = 1

2CV 2 = 12Q2

C = stored energy.

E = Q/Aǫ0

= electric field strength very near a charged conductor.

Chapter 18 Equations

Electric current:

I = ∆Q∆t , or ∆Q = I∆t, definition of current.

V = IR, or I = V/R, Ohm’s law.

R = ρLA = calculation of resistance.

ρT = ρ0[1 + α(T − T0)] = temperature-dependent resistivity.

Electric power:

P = IV , P = I2R, P = V 2/R, P = instantaneous energy/time.

Alternating current:

V = V0 sin 2πft = time-dependent AC voltage. I = I0 sin 2πft = time-dependent AC current.

Vrms =√

V 2 = V0/√2 = root-mean-square voltage. Irms =

√

I2 = I0/√2 = root-mean-square current.

AC power in resistors:

P = 12I

20R = 1

2V20 /R = 1

2I0V0 = average power. P = I2rmsR = V 2rms/R = IrmsVrms = average power.

Eq.-2

V = k Qr = potential produced by a point charge or outside a spherical charge distribution.

PE = qV = potential energy for a test charge at a point in a field.PE = k Q1Q2

r12= potential energy of a pair of charges.

Capacitance:Q = CV , C = Kε0

Ad = capacitor equations.

U = 12QV = 1

2CV 2 = 12

Q2

C = stored energy.

E = Q/Aε0

= electric field strength very near a charged conductor.

Chapter 18 Equations

Electric current and power:I = ∆Q

∆t , ∆Q = I∆t current definition. V = IR, I = V/R Ohm’s law.R = ρL/A calculation of resistance. ρ = ρ0[1 + α(T − T0)] resistivity changes.P = IV , P = I2R, P = V 2/R. P = instantaneous work/time.

Alternating current:V = V0 sin 2πft = time-dependent AC voltage. I = I0 sin 2πft = time-dependent AC current.

Vrms =√

V 2 = V0/√

2 = root-mean-square voltage. Irms =√

I2 = I0/√

2 = root-mean-square current.AC power in resistors:

P = 12I2

0R = 12V 2

0 /R = 12I0V0 = average power. P = I2

rmsR = V 2rms/R = IrmsVrms = average power.

Chapter 19 Equations

Resistor CombinationsReq = R1 + R2 + R3 + ... (series) 1

Req= 1

R1+ 1

R2+ 1

R3+ ... (parallel)

Real batteriesVab = E − Ir (terminal voltage) Vab = IR (connected to load R)

Kirchhoff’s Rules∑∆V = 0 (loop rule, energy conservation)

∑I = 0 (node rule, charge conservation)

Chapter 20 Equations

Magnetic forces, torqueF = IlB sin θ (on a current) F = qvB sin θ (on a moving charge)F/l = µ0

2πI1I2

d (between currents) F = qvB = mv2/r (during cyclotron motion)τ = NBAI sin θ (torque on a coil) v = ωr = 2πfr = 2πr/T (circular motion)

Magnetic FieldsB = µ0

2πIr (due to long straight wire) B = µ0IN/l (inside a solenoid)

Right Hand RulesForce (thumb) = [I (4 fingers)] × [magnetic field (palm)] (force on a current)Force (thumb) = [qv (4 fingers)] × [magnetic field (palm)] (force on a moving charge)Current (thumb) ⇐⇒ [magnetic field (4 fingers)] (magnetic field around a wire)Current (4 fingers) ⇐⇒ [magnetic field (thumb)] (magnetic field inside a current loop)

Chapter 21 Equations

Faraday’s Induced EMFΦB = BA cos θ (magnetic flux) E = −N ∆ΦB

∆t (induced emf)E = Blv (moving conductor) E = NBAω sinωt (AC generator)V − E = IR (motor’s counter-emf) E1 = −M ∆I2

∆t (mutual inductance emf)VS/VP = NS/NP (transformer equation) IP VP = ISVS (power in = power out)

Eq.-2

Prefixes

a=10−18, f=10−15, p=10−12, n=10−9, µ = 10−6, m=10−3, c=10−2, k=103, M=106, G=109, T=1012, P=1015

Physical Constants

k = 1/4πε0 = 8.988 GNm2/C2 (Coulomb’s Law) ε0 = 1/4πk = 8.854 pF/m (permittivity of space)e = 1.602× 10−19 C (proton charge) µ0 = 4π × 10−7 T·m/A (permeability of space)me = 9.11× 10−31 kg (electron mass) mp = 1.67× 10−27 kg (proton mass)c = 3.00× 108 m/s (speed of light) c = 2.99792458× 108 m/s (exact value in vacuum)

Units

NA = 6.02× 1023/mole (Avogadro’s #) 1 u = 1 g/NA = 1.6605× 10−27 kg (mass unit)1.0 eV = 1.602× 10−19 J (electron-volt) 1 V = 1 J/C = 1 volt = 1 joule/coulomb1 F = 1 C/V = 1 farad = 1 C2/J 1 H = 1 V·s/A = 1 henry = 1 J/A2

1 A = 1 C/s = 1 ampere = 1 coulomb/second 1 Ω = 1 V/A = 1 ohm = 1 J·s/C2

1 T = 1 N/A·m = 1 tesla = 1 newton/ampere·meter 1 G = 10−4 T = 1 gauss = 10−4 tesla

Chapter 22 Equations

Electromagnetic waves:

| #E|/| #B| = c = 1/√ε0µ0, (fields) fλ = c (wave equation)

Energy density, intensity, power:

u = ε0E2 = B2

µ0(instantaneous energy density) u = 1

2ε0E20 = B2

02µ0

(average energy density)

I = uc = 12ε0E

20c (EM waves intensity) I = P/A = P/(4πr2) (intensity definition)

Chapter 23 Equations

Reflection, Mirrors:

θr = θi (angle of incidence = angle of reflection) f = r/2 (focal length of spherical mirror)1do

+ 1di

= 1f (mirror equation) m = −di/do = hi/ho (linear magnification)

di > 0 ⇒ real, light side. di < 0 ⇒ virtual, dark side.

m > 0 ⇒ upright. m < 0 ⇒ inverted.

|m| > 1 ⇒ magnified. |m| < 1 ⇒ diminished.

Refraction, Lenses:

n = c/v (index of refraction) n1 sin θ1 = n2 sin θ2 (Snell’s Law)1do

+ 1di

= 1f (lens equation) m = −di/do = hi/ho (linear magnification)

di > 0 ⇒ real image, light (opp.) side. di < 0 ⇒ virtual image, dark (same) side.

m > 0 ⇒ upright. m < 0 ⇒ inverted.

|m| > 1 ⇒ magnified. |m| < 1 ⇒ diminished.

Eq.-1

Chapter 24 Equations

Wave properties, interference:

λn = λvacuum/n (wavelength in a medium) ∆x = d sin θ (path difference in double slits)

d sin θ = mλ (double slits bright fringes) d sin θ = (m+ 1/2)λ (double slits dark fringes)

Diffraction:

D sin θ = mλ (single slit minima) y = L tan θ (position on a screen)

d sin θ = mλ (diffraction grating maxima) d = 1/(lines per meter).

Thin film interference:

∆x = path 1© - path 2© (path difference) reflect from higher n ⇒ extra λ/2 path change.

∆x = mλ (constructive interference) ∆x = (m+ 1/2)λ (destructive interference)

Polarization:

I = I0 cos2 θ (transmission thru polarizer) I = 12I0 (transmission of unpolarized light)

Chapter 25 Equations

Cameras

f/D = f-number, or lens aperture film exposure = exposure time / f-number.1do

+ 1di

= 1f (lens equation) m = −di/do = hi/ho (image size and magnification)

Lens power

P = 1/f (power in diopters).

Vision correction

Far point FP = ∞. (good vision) Near point = NP ≤ 25 cm. (good vision)

Nearsighted. Use lens to get FP=∞. Farsighted. Use lens to get NP=25 cm.

Simple magnifier

θ = h0NP (angular size viewed at NP.) θ′ = h0

d0(angular size viewed at any d0.)

M = θ′θ = NP

do(ang. Mag. viewed at any d0.) M = θ′

θ = NPf (ang. Mag. viewed at d0 = f .)

Telescopes

M = θ′θ = fobj

feye(angular magnification)

Eq.-2

Prefixesa=10−18, f=10−15, p=10−12, n=10−9, µ = 10−6, m=10−3, c=10−2, k=103, M=106, G=109, T=1012, P=1015

Physical Constants

k = 1/4πε0 = 8.988 GN·m2/C2 (Coulomb’s Law) ε0 = 1/4πk = 8.854 pF/m (permittivity of space)e = 1.602× 10−19 C (proton charge) µ0 = 4π × 10−7 T·m/A (permeability of space)me = 9.11× 10−31 kg (electron mass) mp = 1.67× 10−27 kg (proton mass)c = 3.00× 108 m/s (speed of light) c = 2.99792458× 108 m/s (exact value in vacuum)h = 6.62607× 10−34 J·s (Planck’s constant) h = 1.05457× 10−34 J·s (Planck’s constant/2π)σ = 5.67× 10−8 W/(m2·K4) (Stefan-Boltzmann const.) hc =1239.84 eV·nm (photon energy constant)

Units

NA = 6.02× 1023/mole (Avogadro’s #) 1 u = 1 g/NA = 1.6605× 10−27 kg (mass unit)1.0 eV = 1.602× 10−19 J (electron-volt) 1 V = 1 J/C = 1 volt = 1 joule/coulomb1 F = 1 C/V = 1 farad = 1 C2/J 1 H = 1 V·s/A = 1 henry = 1 J/A2

1 A = 1 C/s = 1 ampere = 1 coulomb/second 1 Ω = 1 V/A = 1 ohm = 1 J·s/C2

1 T = 1 N/A·m = 1 tesla = 1 newton/ampere·meter 1 G = 10−4 T = 1 gauss = 10−4 tesla

Chapter 22 Equations

Electromagnetic waves:| $E|/| $B| = c = 1/√ε0µ0, (fields) fλ = c (wave equation)

Approximate wavelengths λ for types of EM waves:0 (γ-rays) 30 pm (x-rays) 3 nm (uv) 400 nm (visible) 700 nm (ir) 300 µm (µ-waves) 3 cm (radio) ∞

−→ −→ −→ increasing wavelength −→ −→ −→

Chapter 26 Equations

Time dilation and length contraction:∆t = γ∆t0 = ∆t0/

√1− v2/c2 L = L0/γ = L0

√1− v2/c2

γ = 1/√

1− v2/c2 (relativistic factor) v/c =√

1− 1/γ2 (velocity)

Dyanmics, mass, energy:p = γm0v (relativistic momentum) mrel = γm0 (relativistic mass)E0 = m0c2 (rest energy) E = γm0c2 = mrelc2 (relativistic energy)KE = E −E0 = (γ− 1)m0c2 (kinetic energy) E = E0 + KE =

√p2c2 + m2c4 (relativistic energy)

Eq.-1

Chapter 27 Equations

Blackbody radiation, photons, photo-electric effect:λpT = 2.90 mm·K (Wien’s Law) I = σT 4 (intensity or power/area)E = nhf, n = 1, 2, 3... (quantized radiation energy) E = hc/λ = (1240 eV · nm)/λ (photons)E = hf = W0 + KEmax (photo-electrons) hc/λmax = W0 (work function)KEmax = eV0 (stopping potential) vmax =

√2KEmax/m (max. speed)

Momentum, matter waves:p = h/λ (photon momentum) λ′ = λ + h

mc (1− cos φ) (Compton effect)λ = h/p = h/(mv) (de Broglie wavelength) v =

√2q∆V/m (acceleration thru potential)

Bohr Model:hf = En − En′ (quantum jump) L = mvr = n h

2π (Bohr’s quantization)

rn = n2

Z r1 (Bohr radii) r1 = h2

4π2mke2 = 52.9 pm (1st Bohr radius)

En = −(13.6 eV)Z2

n2 (Bohr energies) En = 12mv2 − kZe2

rn(total energy)

n = 1, 2, 3, ... (Bohr’s quantum number)

Chapter 28 Equations

Wave functions:N ∝ I ∝ | $E|2 (photon detection) N ∝ |Ψ|2 (particle detection)

Heisenberg Uncertainty Principle:∆x ∆px ≥ h (uncertainty principle) h = h

2π = 1.05459× 10−34 J·s∆E ∆t ≥ h (energy-time form) (← These are approximate relations.)∆E = ∆m · c2 (Einstein’s mass-energy equivalence) (← This one is exact.)

Quantum numbers for atoms:principle quantum number n = 0, 1, 2, 3... En = −(13.6 eV)/n2 (energy of hydrogen states)orbital quantum number l = 0, 1, 2...(n− 1) L =

√l(l + 1) h (angular momentum mag.)

magnetic quantum number ml = −l to + l Lz = mlh (z-component of $L)spin quantum number ms = − 1

2 , + 12 Sz = msh (z-comp., spin angular momentum)

shell means a particular value of (n) is given. sub-shell means values of (n, l) are given.orbital means particular (n, l,ml) are given. state means particular (n, l,ml,ms) are given.l = 0, 1, 2, 3, 4, 5, 6... are indicated with respective letters: s, p, d, f, g, h,...Pauli exclusion principle: No two electrons in an atom can occupy the same quantum state.Subshells in order of increasing energy: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p(They fill in order of increasing n + l, or increasing n if there is a tie.)

Eq.-2

Prefixes

a=10−18, f=10−15, p=10−12, n=10−9, µ = 10−6, m=10−3, c=10−2, k=103, M=106, G=109, T=1012, P=1015

Physical Constants

k = 1/4πε0 = 8.988 GN·m2/C2 (Coulomb’s Law) ε0 = 1/4πk = 8.854 pF/m (permittivity of space)e = 1.602× 10−19 C (proton charge) µ0 = 4π × 10−7 T·m/A (permeability of space)c = 3.00× 108 m/s (speed of light) c = 2.99792458× 108 m/s (exact value in vacuum)me = 9.1094× 10−31 kg = (electron mass) mp = 1.67262× 10−27 kg = (proton mass)mn = 1.67493× 10−27 kg = (neutron mass) hc = 1240 eV·nm (photon energy = hc/λ)h = 6.6262× 10−34 J·s (Planck’s constant) h = 1.05459× 10−34 J·s (Planck’s constant/2π)

Units

NA = 6.02× 1023/mole (Avogadro’s #) 1 u = 1 g/NA = 1.6605× 10−27 kg = 931.5 MeV/c2

1.0 eV = 1.602× 10−19 J (electron-volt) 1 V = 1 volt = 1 J/C = 1 joule/coulomb1 F = 1 farad = 1 C/V = 1 C2/J 1 H = 1 henry = 1 V·s/A = 1 J/A2

1 A = 1 ampere = 1 C/s = 1 coulomb/second 1 Ω = 1 ohm = 1 V/A = 1 J·s/C2

1 T = 1 tesla = 1 N/A·m = 1 newton/ampere·meter 1 G = 1 gauss = 10−4 T = 10−4 tesla1 Bq = 1 becquerel = 1 decay/s 1 Ci = 1 curie = 3.70× 1010 decays/s = 37.0 GBq

Some Masses (for neutral atoms)

electron = 0−1e = 0.00054858 u = 0.51100 MeV/c2 proton = 11p = p =1.007276 u = 938.27 MeV/c2

neutron = 10n = n = 1.008665 u = 939.57 MeV/c2 hydrogen = 1

1H = 1.007825 u = 938.78 MeV/c2

deuterium = 21H = d = 2.014102 u tritium = 3

1H = t = 3.016049 uhelium-3 = 3

2He = 3.016029 u helium-4 = 42He = α = 4.002603 u

Chapter 30 Equations

Nuclides:

A = N + Z, (mass, neutron, proton numbers) r = (1.2 fm) A1/3 (nuclear radius)

∆E =[(mass of parts) - (mass of nuclide)]c2 (binding energy)

Q = [Mparent −Mproducts]c2 (disintegration energy)

Half-life and decay constant

N = N0e−λt (decay of parent nuclei)∣∣∆N∆t

∣∣ = Nλ (radio-activity)

t = −1λ ln(N/N0) (time when N nuclei remain) M = Nm = mass = (# of nuclei)×(nuclear mass)

λT 12= ln 2 (decay constant, half-life) τ = 1

λ (mean life time)

#(146C)/#(126C) = 1.2× 10−12 (live carbon ratio) 1 year = 3.156× 107 seconds

Chapter 31 Equations

Reactions:

Q = [Mreactants −Mproducts]c2 (reaction energy)

Q > 0 (Q = mass converted to energy) Q < 0 (|Q| = threshold energy)

Energy, power and mass in nuclear reactors:

E = mc2 (Einstein’s mass-energy equivalence) P = E/t (power)

E = NQ [energy= (# of reactions)×(reaction energy)] 1 u · c2 = 931.5 MeV

M = Nm [mass used= (# of reactions)×(reaction mass)]

Eout = eEin [output energy = (efficiency)×(input energy)]

Eq.-1