Question

homework due soon !!

Question 3 25 pts A current is spread uniformly over the cross sectional area of a hollow infinite cylindrical shell with inner radius a and outer radius b. Find the magnitude of the magnetic field in the regions a) r<a (5 pts) b) a <r<b (15 pts) c) b<r (5 pts) I O a b

Answer 1

Current density, J = I / ($\pi$b² - $\pi$ a²)

a) r < a
Using Ampere's law,
B * 2$\pi$r = $\mu$ o * I_enc
Where B is the magnetic field, r is the distance from the center, $\mu$ o is the vacuum permeability, and I_enc is the enclosed current.
Here I_enc = 0
B = 0

b) a < r < b
Here I_enc = J * ($\pi$r² - $\pi$ a²)
= [I / ($\pi$b² - $\pi$ a²)] * ($\pi$r² - $\pi$ a²)
= I * (r² - a²) / (b² - a²)

Using Ampere's law,
B * 2$\pi$r = $\mu$ o * I_enc
Substituting I_enc,
B = ($\mu$o * I) / (2$\pi$r) * [(r² - a²) / (b² - a²)]

c) b < r

Here I_enc = I

Using Ampere's law,
B * 2$\pi$r = $\mu$ o * I_enc
Substituting I_enc,
B = ($\mu$o * I) / (2$\pi$r)