Question

A long wire carries a current density proportional to the distance from its center, J=(Jo/ro)•r, where Jo and ro are constants appropriate units. Determine the magnetic field vector inside this wire.

Answer 1

`B = \mu_0((1)/(3) (J_0)/(r_0) r^2)`

Step-by-step explanation:

As the current density is given as

`J = (J_0)/(r_0)r`

now we have current inside wire given as

`i = \int J(2\pi r)dr`

`i = \int (J_0)/(r_0) r(2\pi r)dr`

`i = 2\pi (J_0)/(r_0) \int r^2 dr`

`i = (2)/(3) \pi (J_0)/(r_0) r^3`

Now by Ampere's law we will have

`\int B. dl = \mu_0 i`

`B. (2\pi r) = \mu_0((2)/(3) \pi (J_0)/(r_0) r^3)`

`B = \mu_0((1)/(3) (J_0)/(r_0) r^2)`