Discussion
A point charge \(q\) is performing simple harmonic oscillations of amplitude \(A\) at angular frequency \(\omega\). Using Larmor’s formula, the power radiated by the charge is proportional to
Answer: D
The radiated power is proportional to \(q^{2}\omega^{4}A^{2}\).
Larmor's formula gives the power radiated by an accelerating charge as \(P = \frac{q^{2}a^{2}}{6\pi\epsilon_{0}c^{3}}\), proportional to the square of both the charge and the acceleration. For simple harmonic motion \(x = A\cos\omega t\), the acceleration amplitude is \(\omega^{2}A\), so
\[ P \propto q^{2}(\omega^{2}A)^{2} = q^{2}\omega^{4}A^{2} \]
The options with a single power of \(q\) forget that power goes as the field squared, and those with \(\omega^{2}\) use the velocity amplitude \(\omega A\) instead of the acceleration.
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