Larmor Radiation

Worked Example

Consider a particle with charge q = e (1.602176634 × 10^-19 C) undergoing an accelerationa = 10¹⁵ m/s². The total radiated power P is given by the Larmor formula:

P = (q^2 * a^2) / (6 * π * \epsilon_0 * c^3)

Using the exact SI constants c = 299792458 m/s and ε₀ = 8.8541878128 × 10^-12 F/m, the radiated power is approximately 5.708326765029508 × 10^-24 W.

The radiation is not isotropic. The power radiated per unit solid angle depends on the polar angle θ from the acceleration axis:

dP/dΩ = (3 * P / 8π) * sin^2(θ)

At θ = 90°, the power per solid angle is maximized at approximately 6.813813160786607 × 10^-25 W/sr.

Experiment

You can interactively change the charge multiples, acceleration scale, and measurement polar angle in the module to see how they affect the total power and the angular distribution of radiated power. The interactive tool is available at:
https://riano.app/module/larmor-radiation.

1. Change the Charge multiples to -2 to see how the total radiated power increases by a factor of 4.
2. Change the Polar angle to 0° and observe that the power per solid angle drops to zero, matching the expected radiation lobe shape.

Limitations

This module assumes the particle velocity is strictly nonrelativistic (v ≪ c). The Larmor formula does not apply to relativistic particles; for those, the Liénard-Wiechert potentials and Liénard's generalization must be used. Furthermore, the calculation does not include radiation reaction (the Abraham-Lorentz force) or solve the full trajectory over time.

References

The formulation and angular distribution of nonrelativistic radiation can be reviewed in detail via the following external reference.

• Larmor Formula - The University of Texas at Austin