A long hollow non-conducting cylinder of radius 0.060 m and length 0.70 m carries a uniform charge per unit area of 4.0 C/m^2 on its surface. Beginning from rest, an externally applied torque causes the cylinder to rotate at constant acceleration of 40 rad/s^2 about the cylinder axis. Find the net power entering the interior volume of the cylinder from the surrounding electromagnetic fields at the instant the angular velocity reaches 200 rad/s.

You need to use the equation for the power radiated by accelerating charge. This requires fairly advanced E&M "ed potential" theory.

The total charge on the cylinder surface is
Q = 2*pi*R*L*(4.0 C/m^2) = 1.056 C

The charge accelerates at a rate
a = R*w^2 = 2400 m/s^2

The radiated power (into the cylinder, to keep it accelerating) is
P = (2/3)*k Q^2*a^2/c^3,

where k is the Coulomb constant, 8.99*10^9 N/m^2/C^2 and c is the speed of light.
(Ref.: Reitz and Milford, Foundations of Electromagnetic Theory)
This is a nonrelativistic formula, requiring
w*R/c <<1
I get 1.4*10^-9 Watts

But the answer is 4.6 micro watts

Anyhow thanks for your time

The surface charge has both centripetal and tangential acceleration, but the latter is negligible. I did not include it. I cannot explain the large discrepancy. See what you get using the formula for radiation by accelerating charge.

To find the net power entering the interior volume of the cylinder, we need to calculate the rate at which work is done on the cylinder by the external torque as it rotates.

First, let's find the moment of inertia of the cylinder. The moment of inertia of a hollow cylinder rotating about its axis is given by the formula:

I = (1/2) * M * (R1^2 + R2^2)

where M is the mass of the cylinder and R1, R2 are the inner and outer radii of the cylinder. Since the cylinder is non-conducting and hollow, its mass can be calculated using the surface charge density and the length of the cylinder:

Mass of the cylinder = (surface charge density) * (area)

The surface area of the cylinder can be calculated as the difference between the areas of the outer and inner cylinders:

Area = π * (R2^2 - R1^2)

Substituting the given values, we have:

Mass of the cylinder = (4.0 C/m^2) * π * (0.7 m) * (π * (0.060 m)^2)

Next, we need to find the torque acting on the cylinder. The torque is given by:

τ = I * α

where α is the constant angular acceleration and I is the moment of inertia of the cylinder. Given that α = 40 rad/s^2, and we have calculated I using the previous formula, we can find the torque.

Now, let's calculate the work done on the cylinder as it reaches an angular velocity of 200 rad/s. The work done is given by the formula:

Work = (1/2) * I * (ω^2 - ω0^2)

where ω and ω0 are the final and initial angular velocities, respectively. Substituting the given values, we have:

Work = (1/2) * I * [(200 rad/s)^2 - (0 rad/s)^2]

Finally, to find the power, we need to divide the work done by the time taken. Since the problem doesn't provide the time taken, we cannot calculate the net power entering the interior volume of the cylinder without this information.