Radiation from accelerated charge and Larmor formula

The Larmor formula, also known as the Larmor power formula, describes the power radiated by an accelerated charged particle, such as an electron, as it moves in a circular or curved path due to its acceleration. This formula is a fundamental concept in classical electrodynamics and helps us understand how charged particles emit electromagnetic radiation when they undergo acceleration.

Key points about the Larmor formula:

1. Power Radiated: The formula calculates the power radiated by the charged particle as it accelerates. This power is emitted in the form of electromagnetic radiation, typically in the radio or microwave frequency range for most everyday situations involving charged particles.

2. Acceleration Matters: The power radiated is directly proportional to the square of the acceleration of the charged particle. This means that particles undergoing rapid acceleration emit more power as electromagnetic radiation.

3. Speed of Light: The speed of light (c) in the denominator of the formula indicates that the emitted radiation travels at the speed of light. This is a fundamental property of electromagnetic waves.

4. Direction of Acceleration: The Larmor formula applies to charged particles undergoing acceleration in any direction, not just in circular or helical paths. As long as there is acceleration, radiation is emitted.

5. Classical Electrodynamics: It's important to note that the Larmor formula is a classical result and is valid for non-relativistic velocities (i.e., velocities much less than the speed of light). In the realm of special relativity, the description of radiation emitted by accelerated charges becomes more complex and is described by different formulas.

Derivation of Larmor formula

The electric and magnetic field due to accelerated charge

There are two parts of electric field.

The first part is due to velocity and the second part is due to acceleration. Radiation is given by only accelerated part. Let

be accelerated part. For v<< c, 𝛽 is very small.

The part due to radiation is given by poynting vector. According to this vector, energy flux is given by

The energy flux crossing on area element 'da'

where dΩ = da/ R² = solid angle

= energy/ sec = dP

So, the power per solid angle = dP / dΩ

Substituting for Ea

Total power is given by integrating over total solid angle.

This formula is known as Larmor's formula.

In summary, the radiation from an accelerated charge is a general concept in electromagnetism, while the Larmor formula specifically quantifies the power of this radiation. The Larmor formula is a key result in understanding how electromagnetic radiation is generated by accelerated charges and provides insights into the rate at which energy is carried away by this radiation.

The Larmor formula is significant in various fields, including accelerator physics, radiation therapy in medicine, and the understanding of electromagnetic radiation from moving charged particles in astrophysical contexts, such as synchrotron radiation emitted by charged particles in accelerators and celestial objects.

This note is a part of the Physics Repository.