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 = (πœ‡β‚€πœŒπ‘ŸΒ²πœ”) / 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(Ο€rΒ²L)

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

B = (πœ‡β‚€π‘π‘–) / L

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

i = (𝜌(Ο€rΒ²L)Ο‰) / 2Ο€

i = (𝜌rΒ²LΟ‰) / 2

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

B = (πœ‡β‚€πœŒrΒ²Ο‰) / 2

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

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