- #1

Natchanon

- 31

- 3

- Homework Statement
- Note: E, B, A, J and x are vectors, and w is omega, the angular frequency.

Determine the vector potential A(x,t) and scalar potential V(x,t) in the Lorentz gauge, for the linearly polarized plane wave described by

E(x,t) = E_0 e^(i(kz-wt)) x_hat

B(x,t) = B_0 e^(i(kz-wt)) y_hat,

with the boundary condition that the potentials must be finite at infinity. (Hint: Let V = 0)

- Relevant Equations
- 1. E and B above

2. Lorentz gauge: ∇⋅A = -μ_0 ε_0 ∂V/∂t

3. Gauss' law in terms of V: -∇^(2)V + μ_0ε_0∂^(2)V/∂t^(2) = ρ/ε_0

4. Ampere-Maxwell law in terms of A: -∇[SUP]2[/SUP]A + μ[SUB]0[/SUB]ε[SUB]0[/SUB]∂[SUP]2[/SUP]A/∂t[SUP]2[/SUP] = -μ[SUB]0[/SUB]J

5. B_0 = E_0 / c

6. c = w/k

Lorentz gauge:

Gauss's law: -

Ampere-Maxwell equation: -

I started with the hint,

E

mult. both sides by ∂t then integrate to get

Now this looks good to me at first, as it satisfies

Where did I go wrong? Is it my method of finding

**∇**⋅**A**= -μ_{0}ε_{0}∂V/∂tGauss's law: -

**∇**^{2}V + μ_{0}ε_{0}∂^{2}V/∂t^{2}= ρ/ε_{0}Ampere-Maxwell equation: -

**∇**^{2}**A**+ μ_{0}ε_{0}∂^{2}**A**/∂t^{2}= μ_{0}**J**I started with the hint,

**E**= -**∇**V - ∂**A**/∂t and set V = 0, and ended up withE

_{0}e^{i(kz-ωt)}x_hat = - ∂**A**/∂tmult. both sides by ∂t then integrate to get

**A**= -i(E_{0}/ω)e^{i(kz-ωt)}x_hatNow this looks good to me at first, as it satisfies

**B**=**∇**×**A**, which gives the**B**(x,t) equation in the homework statement. And since we let V = 0, it satisfies Gauss' law in vacuum where ρ = 0**.**But when i checked the Ampere-Maxwell equation, setting**J**=**0**because we're in vacuum, the first term is fine as it gives 0, but the second term gives μ_{0}ε_{0}iE_{0}ωe^{i(kz-ωt)}, which isn't equal zero.Where did I go wrong? Is it my method of finding

**A**or do I misunderstand the Ampere-Maxwell equation? Or something else? Is**A**even supposed to be imaginary?