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.

Question

drwls · Answer

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

sand · Answer

But the answer is 4.6 micro watts Anyhow thanks for your time

drwls · Answer

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.