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?

 

[paranormal]

I would approach this problem by first understanding the physical scenario and the equations involved. The question is asking about the magnetic field induced by a time-dependent current through space. This means that the current is changing with time, and the magnetic field will also be changing with time. We can use Maxwell's equations to describe the relationship between the electric and magnetic fields and the current and charge densities.

In this scenario, we have an infinitely long straight line carrying a current which is time-dependent. This means that the current density will also be time-dependent. We can represent this current density as J = I(t)δ(r)z^, where I(t) is the time-dependent current intensity, δ(r) is the 2-D Dirac delta function, and z^ is the unit vector in the z direction.

Using Maxwell's equations in vacuum, we can write the following equations:

div E = ρ/ε0
rot E = -∂t B
div B = 0
rot B = μ0J + μ0ε0∂tE

We can then use the continuity equation, div J = -∂tρ, to simplify the last equation to:

rot B = μ0J

Now, the issue arises with the term μ0J, as the current density J contains the delta function δ(r). To solve this, we can use the fact that the delta function has the property δ'(x) = -δ(x). Applying this property, we get:

rot B = μ0I(t)δ'(r)z^

We can then use the definition of the curl operator, rot A = (∂Ay/∂z - ∂Az/∂y)x^ + (∂Az/∂x - ∂Ax/∂z)y^ + (∂Ax/∂y - ∂Ay/∂x)z^, to get:

rot B = μ0I(t)δ'(r)z^ = μ0I(t)δ'(r)(-y^)

This means that the magnetic field will have a component in the y direction, and its magnitude will be proportional to the current intensity I(t) and the derivative of the delta function δ'(r).

In summary, the magnetic field induced by a time-dependent current through space can be found by using Maxwell's equations and the continuity equation. The presence of the delta function in the current density can