Magnetic field induced by time-dependent current through out space?

[liumylife]

Homework Statement

There's a infinitely-long straight line in vacuum goes through space, carrying a current which is time-dependent. What is the magnetic field induced by the curent?

B, vector, the magnetic field
E, vector, the electric field
J, vector, the current density
ρ, scalar, the charge density
t, time
μ₀, permeability
ε₀, permitivity
div, divergence ( of vector )
rot, curl or rotation ( of vector )
Δ, Laplacian operator
∂_t, partial differential operator with respect to time
r^, θ^ and z^ are unit vectors of r, θ, z

Homework Equations

Maxwell's equations in vacuum:
div E = ρ / ε₀
rot E = -∂_t B
div B = 0
rot B = μ₀ J + μ₀ε₀ ∂_t E

Continuum equation:
div J = -∂_t ρ

The Attempt at a Solution

Take the straight line as z axis, using cylindrical coordinate r, θ, z.
The current density should be J = I(t) δ(r) z^ where the I(t) is the current intensity changing with time, and δ(r) is 2-D Dirac delta function.
Taking the rotation of the last equation in M's equations, I have
-Δ B = μ₀ rot J - (∂_t)² B,
the rot J term brings trouble, what do I get when differentiate delta funtion?

 

[Thoros]

I would use the given current density to determine the vector potential $$\vec A$$
which is related to the B and E fields by:
$$\vec B = \nabla \times \vec A$$$$\vec E = - \frac{\partial \vec A}{\partial t}- \vec \nabla \phi$$

After the gauge choice div A = 0 i got after omitting constants, like c:
$$\square \vec A = \vec j + \vec \nabla (\nabla \vec j)$$

Write this out in your coordinate choice.
I think this can be solved easily for A using Green's functions and doing a Fourier transform. I haven't tried it out myself yet.

If anyone knows a better way, have at it, I'm interested as well.

 

[liumylife]

Solution

I found the solution in Griffiths' book: Chap.10, (10.31), generalization of the Biot-Savart law.

 

[rude man]

Looks like a tough inegration to find A . Easy for a short wire, L << λ or r, but for an infinitely long one?