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
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
Maxwell's equations in vacuum:
div E = ρ / ε0
rot E = -∂t B
div B = 0
rot B = μ0 J + μ0ε0 ∂t E
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?