Phyics B michaelmas: EM

Stokes's theorem

∫F·dr = ∫∫∇×F·dS

Divergence Theorem

∫F·dS = ∫∫∇·F·dV

torque and potential energy of:

a dipole in a uniform electric field

a magnetic dipole in a uniform magnetic field

G = p × E

U = -p·E

G = m × B

U = -m·B

force on:

a dipole in a non-uniform electric field

a magnetic dipole in a non-uniform magnetic field

net force and torque:

electric: F = ∇[p·E]

magnetic: F = ∇ [m · B]

Maxwell's Equations diffrential form

∇·D = ρ_free

∇ · B = 0

∇ × E = − ∂B / ∂t

∇ × H = J_free + ∂D / ∂t

Maxwell's Equations integral form

∮ D · dA = Q_enc

∮ B · dA = 0

∮ E · dl = − d/dt ∫ B · dA

∮ H · dl = I_enc + d/dt ∫ D · dA

potential expression using Green's function, and the four conditions necessary to find G

V(r) = (1/ε₀)∫ρ(r')G(r',r) d³r'

2 BC or 1 IC, G⁻ = G⁺ , dG⁻ /dr - dG⁺ /dr

stored magnetic energy

and density

1/2∫B(r)·H(r)dV

density: 1/2 B(r)·H(r)

Lorentz force law (full)

F = q(E + v × B)

E in terms of V

E = -grad(V)

potential due to a dipole in spherical polar coordinates (far)

V(r, θ) = p cos θ / (4πε₀ · r² )

electric dipole moment?

p = qa

couple G acting on a dipole in a uniform electric field?

G = p × E

potential energy of a dipole in a uniform electric field

U = -p · E

What is the work done dW when increasing the angle θ of a dipole in a uniform field?

dW = |G(θ)|dθ, where G(θ) is the couple acting on the dipole.

force using potential energy

F(r) = -∇U(r)

electric flux

Φ = ∫∫dS · E(r).

continuity equation in electrostatics (conservation of charge)

∂ρ/∂t + ∇·J = 0

E of a uniform sheet of charge?

E = σ/(2ε₀)

E of a uniform line of charge?

E(r) = λ/(2πε₀r)

Poisson's equation

∇²V(r) = -ρ(r)/ε₀

Laplace's equation and when is it applicable

∇²V(r) = 0, applicable when only the surface is charged

types of boundary conditions

Dirichlet: the quantity of interest is specified over a boundary (eg V_surface is given)

Neumann: normal derivative of the quantity of interest is specified over a boundary

Cauchy: mixed

Green's function in free space

G(r, r′) = 1/(4π | r - r′ | )

point source, infinite boundaries

potential between parallel plates

V = Ez

Dipole moment defined through polarisability (α)

p = αE0

voltage and surface charge of a (neutral) conducting sphere in a uniform electric field

V(r) = E₀ cos θ (s³/r² - r)

σ = 3ε₀ E₀ cos θ

E₀ is the external field.

What is the relationship between the radius of a conducting sphere in a uniform field and the strength of the dipole it is equivalent to?

p = 4πε₀s³E₀

What is the capacitance of two parallel cylindrical conductors separated by distance 2D?

C/L = πϵ0 / ln(2D/a)

energy required to assemble a discrete system of charges?

U_N = (1/2) Σ (q_j V_j)

energy stored in a capacitor and inductor

U = (1/2) CV^2

U = (1/2) LI^2

electric potential energy from charge density?

U = (1/2) ∫ d^3r ρ(r)V(r).

electric energy density

UE(r) = 1/2 D(r) · E(r).

force between charged parallel plates?

F = Q²/(2ϵ0A).

Capacitance of Coaxial Cable

C/L = πε₀ / ln(r₁/r₂)

electric displacement

definition

formula in linear ( P proportional to E) dielectrics

in non linear dielectrics

free charge is the source of D, and both free and bound charges are the source of E

linear: D = εε₀E

non linear: D = ε₀E + P

magnetic field in terms of magnetic field strength:

linear and non linear materials

non linear: B = µ0(H + M)

linear: B = µµ₀H

Polarization/ magnetization of a dielectric: definition and formula using χ:

dipole moment(electric/ magnetic) per unit volume

linear: P = ε₀χE , M = χ_m H

where 1 + χ = ε , 1 + χ_m = µ

volume and surface polarization of a dielectric

describes the effective charge:

σ = P · n^

ρ​= −∇⋅P​

capacitance of a capacitor filled with dielectric

C = ε₀εA/d

what component (parallel or perpendicular) of H, B, D, E is conserved across boundaries

D and B : perpendicular

H and E : parallel

electric field INSIDE a dielectric sphere in a uniform field

Ein = 3/(ε+2)E₀

derived by conservation od D across boundary + assuming Ein in unniform + that the external field is the original uniform field plus a dipole field generated by the surface polarisation

Force on current element due to magnetic field.

dF = Idl × B

Biot-Savart law

dB = (µ₀I)/(4πr²) dl × r^

magnetic force between two parallel wires

F = (µ₀I₁I₂)/(2πd)

B on the axis of a current loop

B = (µ₀Ia²)/( 2(a²+x²)^(3/2) )

B in the middle of a long solenoid

µ₀ N/D I

Magnetic dipole moment

m = I∫dS

Magnetic scalar potential

H(r) = −∇φm(r)

Magnetic scalar potential of a current loop

dφ using magnetic dipole

φ using solid angle

dφ = |dm|cosθ /4πr²

φ = IΩ/4π

where the solid angle Ω = dS·r/r³ of a sphere

Ampere's law

∮dl · B = µ₀I = µ₀∫dS·J

magnetic field for a long wire

B(r) = µ₀I/ 2πr

B defined with Magnetic vector potential

B(r) = ∇ × A(r)

Poisson equation for magnetic vector potential A

−∇²A = µ₀J

Ohm's law

resistance with conductivity

J = σE

R = l / Sσ

dipole moment and Magnetisation relationship

m= ∫ M dV

current density:

Magnetisation current density

Magnetisation surface current density

Ordinary (free) electrical current density

J_m = ∇ × M (quivalent currents that reproduce the correct magnetic field, no charge carrying)

J_s​=M×n^ (Appears as a current flowing along the surface)

I=∫​J⋅dS​ (actual motion of free charges)

magnetic field strength in a sphere in a uniform magnetic field

H_in = 3H₀/(µ + 2)

B_ gap in a toroidal core: how to derive it and what is the final answer

use B_gap = B_in since the perpendicular component is conserved, ampere's law on the loop, and B_gap = µ₀H_gap.

B_gap = µ₀NI / l

Faraday's law of magnetic induction

ε = − d/dt ∫ dS · B(r) = − dΦ/dt

self inductance

L = Φ/I

Self-inductance of long solenoid

n²LSµ₀

Self-inductance of coaxial cylinders: formula or how to derive it

µ0L/ 2π ln(b/a)

derive by constructing a surface from the center to the edge and finding the flux through it

Self-inductance of a pair of wires: formula or how to derive it

µ0l/π ln (2D/a )

2D is the distance between the wires

derive by constructing a rectangular loop between the centers.

how are sparks across switches created

V_gap​=L dI/dt​

where V is a voltage across the switch. When the switch is opened dI/dt​ goes to very large (negative) numbers and the voltage across the switch grows. Derived by applaying faraday's law to a current loop.

Mutual inductance

M = M21 = Φ2/I1 = M12 = Φ1/I2

total energy stored using mutual and self inductance

U = 1/2 I₁² L₁ + 1/2 I₂² L₂ +1/2 I₁ I₂ M

coupling coefficient

M = k(L₁ L₂)½

ideal transformer coupling coefficient and voltage ratio

k=1

V2/V1 = N2/N1

Load impedance ratio across a transformer: full expression/ how to derive it

Z₁ = [ jωL₁Z₂(N₁/N₂)² ]/ [ jωL₁+ Z₂(N₁/N₂)² ]

same as jωL₁ in parallel with Z₂(N₁/N₂)²

typically: Z₁ ≈ Z₂(N₁/N₂)²

derived by M = (L₁ L₂)½ combined with the expressions for V1 and V2 from faraday's law and complex representation.

Z1 is the equivalent input impedance seen by the source​

Magnetic energy of discrete circuits

U = 1/2 sum Φ_i I_i

speed of light (in terms of free space permeability/permittivity)

c = 1/ √(ε₀µ₀)

refractive index

n=c/v = √(εµ)

(in non magnetic materials: √ε )

k in terms of wavelength and frequency

k = 2π/λ = ω/c

impedance of free space

in terms of permeability and permittivity

numerical value

in term of electric and magnetic field componenets

Z₀ = √(µ₀/ε₀) = Ex / Hy = 377Ω

relationship between Ex and By in free space wave propagating in the z direction

Ex = cBy

relationship between H₀ and E₀ in plane waves using k

H₀ = 1/Z₀ k^ × E₀

k^ is the unit vector pointing in the direction of propagation

Poynting vector:

formula

what does it signify

imaginary and real component sugnificance

N(r) = E(r) × H(r)

The magnitude of N gives the power flow per unit area, and the direction of N gives the direction in which the power is flowing. Can be complex if H and E are out of phase

real: instantaneous power flow per unit area at a given point

imaginary: time-averaged power flow per unit area at a point

Radiation pressure and how to derive it

R = N/c

Consider now the radiation incident normally on area A of a surface in time dt, In terms of the Poynting vector N. Find energy density. Dividing by the volume per photon the energy density is related to the radiation momentum density g by U = |g|c. radiation pressure is the rate of change of momentum per unit area.

Complex power

P = 1/2 I V∗

(half because of time average)

Snell's law

see image

power Reflection coefficient in terms of

refractive index

impedance

R = ( (n₂ − n₁)/(n₂ + n₁) )²

Γ = ( (Z₂ − Z₁)/(Z₂ + Z₁) )²

relative permittivity in plasma using the plasma frequency and how is the plasma frequency derived

ε = 1 - ω²_p/ω²

where ω²_p = Ne²/(ε₀ m_e)

derived from:

m_e d²r/dt² = −e (E + v × B)

p = -er

N p = P = ε₀χE

Effective dielectric constant in conducting media

ε' = ε + iσ/ωε₀

where ε is negligable in metals where σ is large

skin depth in conducting media:

what is it

formula / how to derive it

relation to the wavevector

The distance into a metal at which an alternating E falls to E₀/e

δ = √(2/σωµ₀µ)

derive by: n = √(ε'µ) , where the real term in ε' can be neglected. k = ω/(c/n) = ( 1 + i )/ δ by definition.

k = ( 1 + i )/ δ

The effective resistance per unit length of a wire considering skin effect

R/L = 1/2πaδσ

effectively the current is assumed to flow uniformly in a thin shell of thickness δ

characteristic impedance of the transmission line

Z = √(L/C)

Voltage reflection and transmission coefficient on a transmission line

r = V2/V1 = (Zt − Z)/(Zt + Z)

t = Vt/V1 = 2Zt/(Zt + Z)

Zt is the load impedance, Z is the line impedance

effective Zin/Z of a transimission line terminated with an impedance Zt, and source impedance = transmittion line impedance = Z , or how to derive it

Zin / Z = (Zt cos ka + iZ sin ka)/ (Z cos ka + iZt sin ka)

derive using the formula for r, and the definition Zin = (Vi+Vr)/(Ii+Ir) at distance z=-a, sub the expression for r, expand to cos and sin

Quarter-wavelength line Zin formula

Zin = Z²/ Zt

Waveguide equation

where k²_g = ω²/c² - m²π²/a² - n²π²/b² = k²₀ -k²_c

where K_c is the cut-off frequency below which we'll get evanescent waves

phase velocity and group velocity in a Waveguide

v_p = ω / k_g

v_g = dω/dk

for vacuum: v_g* v_p = c^2