Constants

Speed of light in vacuum c 2.9979^.10⁸m/s

Elementary charge e 1.602^.10^-19C

Electron mass me 9.109^.10^-31kg

Proton mass m_p 1.673^.10^-27kg

Permittivity constant eo 8.854^.10^-12 F/m

Permeability constant µo 4p^.10^-7H/m
 
 

Coulomb’s law (Fundamental Form):

Electric field

Electric field of a point charge q

Coulomb’s law (General form):

Gauss's law:

Field due to an infinite line charge (charge density l):

.

Electric potential:

General form of the electric potential (if V(infinity) = 0):

Potential from a point charge q:

Work done on a charge Q moved from point r1 to r2 :
W = Q ^. ( V(r2) - V (r1) )

Equations inside a conductor (if NO currents are flowing):

All charges and fields are on the surfaces only. E is normal to the
surface and E different from 0 only outside, with magnitude |E| = s / e₀.

Capacitance:
C = Q / V

Energy stored in Capacitor:

Capacitance of a parallel plate capacitor:

Field between plates:

Linear Dielectrics

Combinations of Capacitors
Capacitance of n capacitors, all parallel:

C = C₁ + C₂ + ... + C_n

Capacitance of n capacitors, all in series:

Dipoles

Force on a dipole:

Energy of a dipole in an electric field: 

Current

Total charge (including sign) going through a cross section per unit time

Current density J: 

(Linear) Resistivity:

Resistance

R=V/I (potential drops in direction of current => DV is negative)

Resistance of a piece of wire of length L and constant cross section A:

R = rL/A.

Resistance of n resistors, all in series:

R = R₁ + R₂ + ... + R_n

Resistance of n resistors, all parallel:

Power dissipated in a resistor: P = RI² = V²/R

EMF (of a battery or circuit):

Work done per unit charge by non-electrostatic forces.

Terminal voltage of a battery: V = EMF - r_internalI (potential increases going from negative to positive terminal => DV is positive).

Current in a single loop with total resistance R: I = EMF/R

Power: P=EMF^.I

Kirchhoff’s Laws:

1) Following any closed loop in a consistent direction (sense), all potential differences have to add up to zero.

2) All currents going into a point, minus all currents flowing out, must add up to zero.

Resistance-Capacitance Circuits:

Total resistance R, total capacitance C:

I(t) = I_o e^-t/t ; t = RC (charging up; I_o = EMF/R)
 
 

Lorentz Force

Cyclotron motion

pperp = qBR (momentum perpendicular to the magnetic field B for a particle with charge q, orbiting on a circular trajectory with radius R);
w = (q/m) . B (angular velocity of a particle with charge q, mass m in a field B).

Force on a wire

Torque on a wire loop enclosing some area A:

Biot-Savart law for a steady current:

Ampere's law in integral form

Special cases:

a) Field of infinitely thin, long straight wire:

b) Field of infinitely long solenoid (very closely wound, with n windings per unit length):

Magnetization

M = average magnetic dipole moment per volume.
 
 
 
 

Linear Media

Define  is the magnetic field due to external currents alone.

Ferromagnetic material

Ignoring remnant magnetization, ferromagnetic behavior can be described by a permeability µ that is a function of the field H: 

µ is large for small H and becomes smaller when M approaches saturation; in the limit of very large fields µ converges against µo .
 
 

Time-varying charges and currents - Electromotive Force

Flux rule for the EMF induced by magnetic fields:

(Motional EMF: Flux change due to change of enclosed surface => EMF due to Lorentz force: ;
Faraday's law: Flux change due to change in field B => induced circular E field:  ).

Lenz’s Law

"A change of magnetic flux through a loop will create an EMF that tends to oppose that change".
 
 
 
 

Displacement current

Ampere’s Law, final version:

Maxwell’s Equations

Mutual Inductance

Two separate wire loops:

(same M = mutual Inductance. Unit: Henry = Volt-seconds/Ampere)

Example: Two concentric solenoids of equal length l and cross sectional area A, with n₁ and n₂ windings per unit length: M = µ_on₁n₂lA

(Self-)Inductance

Single circuit:  (Back EMF).

Example: Long solenoid of length l and cross sectional area A, with n windings per unit length: L = µ_on²lA

R-L circuit:

Total resistance R, total inductance L:

I(t) = I_o (1 - e^-t/t ); t = L/R (B field building up; I_o = EMF/R)

Magnetic Field energy:

L-C circuit:

Damped Oscillator (R-L-C):

AC circuits (series):

EMF(t) = EMF_o cos(wt+f); i(t) = i_o cos(wt) . f is the phase angle.

Kirchhoff’s law: EMF(t) = V₁(t) + V₂(t) + ... ; V_n(t) = V_no cos(wt+f_n).

_{Impedance: Z = EMFo/io .}

(Purely Ohmic) resistance: V_Ro = R i_o ; f_R = 0.

(Pure) inductance: V_Lo = X_L i_o ; X_L = wL; f_L = +90^o (V is ahead of I).

(Pure) capacitance: V_Co = X_C i_o ; X_C = 1/(wC); f_C = -90^o (I is ahead of V).

RLC impedance: _{; Phase angle: ; resonance frequency:} 

_{Power in AC circuits:}

RMS voltage: VRMS=(1/sqrt2) V_peak ; RMS current: I_RMS=(1/sqrt2) I_peak

Power Factor: cosf

Impedance: Z = V_RMS/ I_RMS

Transformers:

Electromagnetic Waves:

A plane wave (with angular frequency w=21f and wave number k=21/l) moving in the positive x-direction (with phase velocity ) and polarized along the y-axis has the form 
 
 

Propagation of Light:

In a medium with index of refraction n, the phase velocity is v_phase = c/n.

Angle of reflection = Angle of incidence (on opposite sides of normal on surface)

Snell’s Law: n_a sinq_{a = nb sin}q_b (for a light ray propagating from medium a to medium b, with angles measured relative to the normal); minimum angle of total internal reflection: sin q_a^min= nb / na.

Intensity of light transmitted through a polarizer: I = I_in cos^₂f, where f is the angle between polarizer axis and incident direction of polarization.

Laws for optical elements:

Definitions: P = object point, s = object distance, P’ = image point,
s’ = image distance (<0 for virtual image), y = object height, y’ = image height (<0 for inverted image), m = y’/y = magnification;
F (F₁ and F₂) = (primary and secondary) focal point(s), f = focal length
(<0 if virtual focus), C = centerpoint (for radius), R = radius of curvature (<0 if centerpoint is on opposite side of outgoing rays), V = vertex (nominal position of center of element).

Laws: m = -s’/s , 1/s + 1/s’ = 1/f (true for all situations, including signs); f = R/2 for mirrors, 1/f = (n-1)(1/R₁ - 1/R₂) for lenses.

How to draw ray diagrams: a) incoming rays passing through or heading for (primary) focus emerge parallel, b) incoming parallel rays emerge heading for (or coming from) the (secondary) focus, c) incoming rays heading for the vertex V continue with same angle d) (Mirrors only) rays going through centerpoint C are reflected upon themselves.

Interference:

Two light rays can interfere if they have a fixed phase relationship. In that case, E_tot = E₁ + E₂ for constructive interference (phase difference
DF = 2n1) and E_tot = E₁ - E₂ for destructive interference (phase difference DF = [2n+1]1). If both rays have the same intensity I₁, then the total intensity is I_tot = 4I₁ cos²(DF/2). Possible contributions to phase difference:

Optical path length differences Db => DF = 21 Db/l (l = l_o/n is the wavelength in the medium). Example: Double slits separated by d. Maxima at outgoing ray angles with sinq_m = ml/d, minima at sinq_m = (m+1/2)l/d. Intensity distribution: I(q) = 4I₁ cos²(1d sinq/l).

Reflection on a surface going from medium with n_a to medium with
n_b > n_a: DF = 1. Example: Reflection on both sides of a thin film on glass for a ray entering from air (n_air<n_film<n_glass). Destructive interference for reflected light (constructive interference for transmitted light) if thickness d=l/4.

Diffraction:

Single slit of width a: Minima (destructive interference) are at
sinq_{m = m}l/a (m <> 0). Intensity distribution: .

Multiple slits: Sharp maxima at sinq_m = ml/d.

Resolving power of a lens or mirror (Raleigh limit): sinq_min = 1.22 l/D
(D = aperture diameter).