How Do Maxwell's Equations Apply to Current Density in an Infinite Cylinder?

[shmiv]

Homework Statement

In an infinite cylinder, there is a current density function of $$\rho$$ with the following expression:

$$J_z(\rho) = 3(\rho-1)$$ for $$\rho \leq 1$$
$$J_z(\rho) = 0$$ for $$\rho > 1$$

Determine the electric and magnetic fields in all space.

Homework Equations

As far as I can tell, this should be solvable with Maxwell's equations in differential form.

The Attempt at a Solution

Even though it's not explicitly stated, I've assumed that the electric current density in the x and y directions to be 0 as a function of $$\rho$$. So far, I've tried using Maxwell's equations to arrive at a solution, but it seems as though there isn't enough information. Am I missing something here? A nudge in the right direction would be much appreciated!

Thanks in advance for your help!

 

[anathema]

Hi there,

You are correct in assuming that the electric current density in the x and y directions is 0, as it is only given as a function of ρ and z. To solve this problem, you will need to use the following equations:

∇ · E = ρ/ε0
∇ · B = 0
∇ x E = - ∂B/∂t
∇ x B = μ0(J + ε0∂E/∂t)

Using these equations, you can find the electric and magnetic fields in the infinite cylinder. Start by finding the electric field using the first equation, and then use the second equation to find the magnetic field. Remember to take into account the different regions where the current density is non-zero and zero.

Once you have the electric and magnetic fields, you can use the last two equations to check your solution and make sure it satisfies Maxwell's equations. Good luck!