Larmor's formula traditionally expresses the total power radiated by an accelerated charge moving at low velocities, but it fails to account for relativistic effects when velocities approach the speed of light. To address this limitation, the formula must be generalized into a form that holds true for relativistic speeds, often referred to as the Liénard radiation formula. The derivation begins by rewriting the classical force equation using Newton's second law, substituting the linear rate of change of momentum into the standard power formula. This process produces a foundational expression for power that sets the stage for introducing four-dimensional vectors in the realm of special relativity.
Transitioning the power equation from non-relativistic to relativistic motion requires substituting classic three-dimensional variables with four-vectors, specifically replacing linear momentum with four-momentum. Additionally, the standard time derivative must be replaced with proper time, which scales according to the Lorentz factor to ensure invariance across different reference frames. Four-momentum is composed of a space part and a time part, where the latter is related to the energy of the particle. By incorporating these definitions, the reformulated power equation becomes dependent on the derivatives of energy and momentum with respect to proper time.
The derivation continues by calculating the time derivatives of relativistic momentum and energy, applying the product rule and chain rule to the Lorentz factor. Relativistic momentum is defined as the product of mass and velocity scaled by the Lorentz factor, while the relativistic energy follows Einstein's mass-energy equivalence similarly adjusted for high speeds. Careful algebraic manipulation reveals that these derivatives are functions of the particle's velocity and acceleration vectors. These components are essential for populating the standardized relativistic power equation, requiring precise tracking of dot and cross products between velocity and its first derivative.
Substituting the expressions for momentum and energy derivatives into the generalized power equation yields the Liénard radiation formula. This final expression captures the total radiated power as a function of the Lorentz factor and the cross product of the particle's velocity and acceleration vectors. The formula accounts for the highly directional nature of radiation emitted by charges moving at near-light speeds. To verify the result, the expression is tested against the non-relativistic case where the velocity is much smaller than the speed of light. The formula correctly reduces back to the original Larmor's formula when these conditions are applied, proving its accuracy as a universal generalization.