Vector and scalar potentials for an EM plane wave in a vacuum

[Natchanon]

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: ∇⋅A = -μ₀ε₀∂V/∂t
Gauss's law: -∇²V + μ₀ε₀∂²V/∂t² = ρ/ε₀
Ampere-Maxwell equation: -∇²A + μ₀ε₀∂²A/∂t² = μ₀J
I started with the hint, E = -∇V - ∂A/∂t and set V = 0, and ended up with
E₀ e^i(kz-ωt) x_hat = - ∂A/∂t
mult. both sides by ∂t then integrate to get A = -i(E₀/ω)e^i(kz-ωt) x_hat
Now 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 μ₀ε₀iE₀ωe^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?

 

[Orodruin]

the first term is fine as it gives 0

It does not. Please show your computation.

Note that the equation is just a wave equation for $\vec A$.

 

[Natchanon]

It does not. Please show your computation.

Note that the equation is just a wave equation for $\vec A$.

Ok, I forgot and treated A as a scalar when calculating ∇^2. So,
∇²A = ∇(∇⋅A) - ∇ × (∇ × A) = ∇(∂A_x/∂x) - (∇ × B)
First term, we have partial A_x / partial x, but x-component of A doesn't depend x, so it's zero, thus, the first term is zero. The second term is μ₀J, which is zero, unless there is something I misunderstand. So even after finding vector laplacian of A, the first term is still 0 for some reason. I don't know what's going on here.

 

[Delta2]

it is $\nabla \times B=\mu_0(J+\epsilon_0\frac{dE}{dt})$ so its not just $\mu_0J$..

(You forgot the displacement current term dE/dt)

 

[Natchanon]

it is $\nabla \times B=\mu_0(J+\epsilon_0\frac{dE}{dt})$ so its not just $\mu_0J$..

(You forgot the displacement current term dE/dt)

Thank you. It all makes sense now!

 

[Orodruin]

Ok, I forgot and treated A as a scalar when calculating ∇^2.

As long as you do things in a Cartesian basis you can factor out the basis vectors and treat the components of $\nabla^2 \vec A$ separately. The issue you were having is solved by #4.