How to Calculate the Magnetic Field Within an Infinitely Long Rotating Metal Cylinder?

What is the formula to find the magnetic field within the cylinder?

Given: An infinitely long metal cylinder rotates about its symmetry axis with an angular velocity omega. The cylinder is charged. The charge density per unit volume is sigma. Find the magnetic field within the cylinder.

Answer:

The magnetic field within the cylinder can be calculated using the formula:

B = (𝜇₀𝜌𝑟²𝜔) / 2

To calculate the magnetic field within the cylinder, we first need to determine the total charge lying in the region. The total charge is given by:

q = 𝜌(πr²L)

Next, we use the formula for the magnetic field inside the cylinder:

B = (𝜇₀𝑁𝑖) / L

Substitute the expression for current 'i' in terms of charge 'q' and angular velocity 'ω':

i = (𝜌(πr²L)ω) / 2π

i = (𝜌r²Lω) / 2

Finally, we obtain the magnetic field within the cylinder by substituting the values into the formula:

B = (𝜇₀𝜌r²ω) / 2

Therefore, the magnetic field within the infinitely long rotating metal cylinder can be calculated using the above formula.