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

1. Mar 9, 2013

liumylife

1. The problem statement, all variables and given/known data
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

2. Relevant equations

Maxwell's equations in vacuum:
div E = ρ / ε0
rot E = -∂t B
div B = 0
rot B = μ0 J + μ0ε0t E

Continuum equation:
div J = -∂t ρ

3. 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?

2. Mar 9, 2013

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.

3. Mar 11, 2013

liumylife

Solution

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

4. Mar 11, 2013

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?