GATE 2022 PH – Question 16

Electromagnetic theory · Poynting vector, Poynting theorem, energy and momentum of electromagnetic waves. · 1 mark · Multiple choice

A charge q oscillates harmonically with amplitude A and angular frequency omega. What is the Larmor radiation power proportional to?

$q\omega^2A^2$
$q\omega^4A^2$
$q^2\omega^2A^2$
$q^2\omega^4A^2$

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Correct answer: (D) $q^2\omega^4A^2$

Explanation

**Larmor formula.** The power radiated by an accelerating charge $q$ is
$$P=\frac{q^2a^2}{6\pi\epsilon_0c^3}\propto q^2a^2 .$$

**Harmonic motion** $x=A\cos\omega t$ has acceleration $a=-\omega^2A\cos\omega t$, so the average of $a^2$ is $\tfrac12\omega^4A^2$:
$$\langle P\rangle\propto q^2\omega^4A^2\quad(\text{option D}).$$