Question

Magnetic vector potential A conductor of radius a carries a uniform current density,

J=J0z.
Show that the magnetic vector potential for rA = −1/4 μ0J0r²z
Hint: It might be easiest for this problem to first find B (or H ).

Answer 1

Using Ampère's Law, we find the magnetic field inside the conductor, and then determine the magnetic vector potential A given the relationship between B and A.

Step-by-step explanation:

The question requires calculating the magnetic vector potential for a conductor with a uniform current density. Using Ampère's Law, we can find the magnetic field (B) created by the current in the conductor. Once B is determined, we can use the relationship between the magnetic vector potential (A) and B (since B is the curl of A) to find the potential.

For a point inside the conductor (r < a), we apply Ampère's law in integral form:

èB • dl = μ0I_enc

where I_enc is the enclosed current. By symmetry, B is tangential and uniform along Ampère's loop. Solving for B, we find:

B = è μ0J0r/2

From the relationship B = ∇ × A, we use the fact that for cylindrical symmetry, A has only a z-component (Az), and knowing the expression for B inside the conductor, we can solve for A using:

∇ × Az = B

Integrating B over the enclosed area gives us the magnetic vector potential A:

Az = -(1/4) μ0J0r^2

This yields the magnetic vector potential for r inside the conductor.