Electromagnetic Radiation

An accelerating charge radiates electromagnetic energy. For non-relativistic motion, the radiated power is given by the Larmor formula. Oscillating electric dipoles are the simplest radiating systems and form the basis of antenna theory.

Key Concepts

Larmor formula: P = q²a²/(6πε₀c³)
Electric dipole radiation: P = ω⁴p₀²/(12πε₀c³)
Radiation pattern: dipole has sin²θ angular dependence
Radiation resistance of antenna: P = I²R_rad
Cyclotron radiation from circular motion in magnetic field

Key Equations

Larmor formula

P = \frac{q^2 a^2}{6\pi\varepsilon_0 c^3}

Dipole radiated power

P = \frac{\mu_0 \omega^4 p_0^2}{12\pi c}

Radiation intensity

\frac{dP}{d\Omega} = \frac{\mu_0 q^2 a^2}{16\pi^2 c}\sin^2\theta

Cyclotron frequency

\omega_c = \frac{qB}{m}

Worked Example

Example Problem

Problem

An electron (q=1.6×10⁻¹⁹ C) has acceleration a=10²⁰ m/s². Find the radiated power using Larmor formula.

Solution

P = q²a²/(6πε₀c³) = (1.6×10⁻¹⁹)²×(10²⁰)²/(6π×8.854×10⁻¹²×(3×10⁸)³) = 2.56×10⁻³⁸×10⁴⁰/(5.093×10¹¹) = 1.52 W.

Practice

Exercises

7 problems

1 of 7

An electron with a=5×10¹⁹ m/s². Find the Larmor radiated power in mW.

Your answer

2 of 7

A dipole antenna (p₀=10⁻²⁹ C·m) oscillates at f=100 MHz (ω=6.28×10⁸ rad/s). Find total radiated power in pW.

Your answer

3 of 7

An electron moves in a circle (r=0.5 cm) in B=0.01 T. Find the cyclotron frequency f_c in GHz.

Unlock Exercise 3