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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.

  • physics -

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

  • physics -

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

  • physics -

    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.

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