Boundary Conditions on Electromagnetic Fields

Question 1

What is the Maxwell-Faraday equation?

Answer 1

∇XE = -dB/dt

Question 2

What are we led to if we integrate the Maxwell-Faraday equation around a closed loop?

Answer 2

Leads to Faraday’s law via Stokes’s theorem.

Question 3

If we have a rectangle with infinitesimally small width and length along the boundary between two media, what does the integral of E.dl equal?

Answer 3

integral of E.dl = (E(2parr)-E(1parr))*L+O(d), where L is the length of the long sides of the box and the last term indicates corrections where d is the depth of the box.

Question 4

What do we assume for the box across the boundary and what equation can we get from this?

Answer 4

Assume that L is so small that the electric field can be assumed to be constant over the loop: (E(1parr)-E(2parr)L = 0Ld*dB(parr)/dt +O(d)

Question 5

Using the equation we found from L being small, what relation can we get?

Answer 5

Take limit of d->0, so can rearrange giving E(2parr) = E(1parr), or n(hat)X(E2-E1) = 0, where n(hat) is a unit vector perpendicular to the boundary

Question 6

What is the equation for the boundary condition on D? What is the integral form?

Answer 6

∇.D = ρ(f) -> closed integral of D.dS = integral of ρ(f) dV = Q(f)

Question 7

If we consider a pillbox straddling a boundary, with surface area A, what equation can we derive for the flux?

Answer 7

closed integral of D.dS = (D(2perp)-D(1perp))A = ρ(f)Ah+σ(f)A, where the first term if a smooth volume distribution of charges and the second term is a surface layer at the boundary of charges.

Question 8

What happens to the pillbox flux equation if we take the limit h->0? What is another way of writing this?

Answer 8

Sub in h=0 and find that (D(2perp)-D(1perp)) = σ(f) or n(hat)*(D2-D1) = σ(f)

Question 9

How can we apply the pillbox model to deduce the boundary condition on B?

Answer 9

Starting with ∇.B = 0, its simple to show that B(2perp)=B(1perp), or n(hat)*(B2-B1) = 0 i.e. the component of the B field perpendicular to the boundary matches on each side, reflecting conservation of magnetic flux.

Question 10

What do we start with to derive the boundary condition on H?

Answer 10

Start with Maxwell-Ampere equation: ∇XH = J(f)+d(t)*D and consider the rectangular loop oriented parallel to the boundary

Question 11

What do we do with the Maxwell-Ampere equation to derive the boundary condition on H?

Answer 11

Can make same assumption as before the d-> for the rectangle, but cant for J because we have to allow for surface current, therefore n(hat)X(H2-H1) = j(f)

Question 12

What is the charge-current boundary condition?

Answer 12

∇.J(f) + d(t)ρ(f) = 0 leads via another pillbox to: n(hat)(J(2f)-J(1f))+dσ(f)/dt = 0

Question 13

What does the charge-current boundary condition imply?

Answer 13

Implies that if the current densities perpendicular to the boundary don’t match, then there must be a change in the charge density on the boundary.

Question 14

What can we approximate the relationship between J and E to?

Answer 14

J(f) = g*E, where g is the conductivity.

Question 15

What can we approximate the relationship between D and E to?

Answer 15

D = εE

Question 16

What is the final charge-current boundary condition equation?

Answer 16

n(hat)(g2E2+ε2dE2/dt - g1E1 - ε1dE1/dt) = 0, which means the flux of gE+εd(t)E is conserved perpendicular to boundaries between conducting media.

Question 17

What is degeneracy?

Answer 17

Sometimes two or more states have the same energy. These states are called degenerate.

Question 18

What is the equation for spin angular momentum?

Answer 18

S = ћsqrt(s(s+1)), where s is the spin quantum number = 0, 1/2, 1, 3/2, …, and Sz = ћm(s), where m(s) is the magnetic spin quantum number = -s, -s+1, …, s-1, s

Question 19

What is the relationship between degeneracy and spin s?

Answer 19

Each energy level of a particle with spin s is 2s+1 degenerate.

Question 20

What does the spin s equal for an electron? What is the degeneracy?

Answer 20

s = 1/2 -> degeneracy = 2s+1 -> 2 spin states (+- 1/2 spin -> splits in B-field)

Question 21

What is the equation for z, accounting for degeneracy?

Answer 21

z = sum over i f g(Ei)*exp(-βEi), where g(Ei) is the number of distinct states associated with energy level i.

Question 22

For a particle with spin s in a quantum well at zero field, what would z equal?

Answer 22

z = sum over i of (2s+1)*exp(-βEi)

Question 23

For a two level system with degeneracies g1 at E=0 and g2 at E = Δ, what does z equal?

Answer 23

z = g1 + g2*exp(-βΔ)

Question 24

What do we do to this equation for z to find an equation for u?

Answer 24

Take ln of both sides and sub in equation for u: find u = g2Δ/(g1*exp(βΔ) + g2)

Question 25

What does the equation for u mean for the two level system example?

Answer 25

At low T, βΔ is large, u-> 0 (ground state). At high T, βΔ &laquo_space;1, u~ g2/(g1+g2) *Δ