Radiation Reaction and Its Puzzles
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You should be comfortable with the Poynting vector, the concept of far-zone radiation fields, and the Larmor formula for the power radiated by an accelerating charge.
Why this matters
Radiation reaction is the mechanical recoil a charge experiences when it emits electromagnetic waves.
When a charge accelerates, it does not merely radiate energy into the distance; it also exerts a force on itself. This self-force, known as radiation reaction, is essential for energy conservation in classical electrodynamics. Without it, a charge could radiate energy indefinitely without losing mechanical energy, violating energy conservation. This lecture explores the derivation of this force, the mathematical difficulties it presents, and the physical puzzles that arise from its non-local nature.
The Physical Intuition
Imagine pushing a heavy object. You apply a force, and the object accelerates. Now imagine that as you push, the object also emits sound waves into the air. The emission of these waves carries away energy. Where does that energy come from? It must come from the work you do. However, there is a subtlety: the object feels a slight resistance due to the emission of the waves. In electromagnetism, this resistance is the radiation reaction force. It is proportional to the rate of change of acceleration, and for oscillating motion it works, on average, against the velocity, damping the motion. This is analogous to air resistance in fluid dynamics, but it is a fundamental property of the electromagnetic field itself, not an interaction with a medium.
The Larmor Formula and Energy Balance
We begin with the result of the previous lecture, the Larmor formula. For a point charge \( q \) with acceleration \( \vec{a} \), the total instantaneous power radiated is given by:
This formula tells us how much energy leaves the system per unit time. To satisfy energy conservation, the mechanical energy of the charge must decrease by exactly this amount. The rate of change of the mechanical kinetic energy \( K = \frac{1}{2}mv^2 \) (in the non-relativistic limit) is given by the work done by the net force \( \vec{F}_{\text{net}} \). If we assume the only force acting is the external force \( \vec{F}_{\text{ext}} \) plus the radiation reaction force \( \vec{F}_{\text{rad}} \), then:
Energy conservation requires that the work done by the external force equals the change in kinetic energy plus the radiated power:
Substituting the expression for \( \frac{dK}{dt} \), we find:
Rearranging terms, we see that the work done by the radiation reaction force must account for the radiated power:
This condition is necessary but not sufficient to determine \( \vec{F}_{\text{rad}} \). It only constrains the component of the force parallel to the velocity. We need a more precise expression.
The Abraham–Lorentz Force
By analyzing the self-field of the charge and integrating the stress tensor over a surface surrounding the charge, one can derive the specific form of the radiation reaction force. In the non-relativistic limit, this force is given by the Abraham–Lorentz formula:
Notice that this force depends on the time derivative of the acceleration, also known as the jerk. This is a crucial feature. The force is not proportional to the acceleration itself, but to its rate of change. This leads to a third-order differential equation of motion, which is mathematically problematic. The standard equation of motion is second-order (Newton's second law), requiring initial position and velocity. A third-order equation requires an initial jerk as well, which is not physically intuitive and leads to the "runaway" solutions discussed later.
We can verify that this force satisfies the energy conservation condition derived earlier. Let us compute \( \vec{F}_{\text{rad}} \cdot \vec{v} \):
Using the identity \( \frac{d}{dt}(\vec{a} \cdot \vec{v}) = \frac{d\vec{a}}{dt} \cdot \vec{v} + \vec{a} \cdot \vec{a} \), we can rewrite the dot product:
Substituting this back into the expression for the power:
The first term is a total time derivative, which integrates to zero over a complete cycle of motion. The second term is:
Thus, the time-averaged work done by the radiation reaction force is indeed equal to the negative of the radiated power, confirming energy conservation.
The Runaway Problem
The Abraham–Lorentz force leads to the equation of motion:
where \( \tau = \frac{\mu_0 q^2}{6\pi m c} \) is a very small time constant (for an electron, \( \tau \approx 6 \times 10^{-24} \) s). This equation can be rearranged to solve for \( \vec{a} \):
If the external force is zero, the equation becomes \( \vec{a} = \tau \frac{d\vec{a}}{dt} \). The general solution is \( \vec{a}(t) = \vec{A} e^{t/\tau} \). This is the "runaway" solution: the acceleration grows exponentially with time, even in the absence of any external force. This is unphysical, as it implies that a free charge will spontaneously accelerate and radiate infinite energy.
The problem arises because the radiation reaction force is not local in time. It depends on the derivative of the acceleration, which couples the motion at time \( t \) to the motion at time \( t + \tau \). This non-locality allows for solutions that do not respect causality in the usual sense. To avoid runaway solutions, one must impose specific boundary conditions, such as requiring that the acceleration vanishes as \( t \to \infty \). This leads to the "pre-acceleration" effect, where the charge begins to accelerate before the external force is applied, which is also physically puzzling.
The derivation
Larmor Formula for Radiated Power
For a point charge \( q \) undergoing acceleration \( \vec{a} \) with speed \( v \ll c \), the radiation (far-zone) part of the Liénard–Wiechert fields falls off as \( 1/R \); integrating the Poynting flux of this field over a large sphere yields the total instantaneous power radiated, \[ P = \frac{q^2 a^2}{6\pi\varepsilon_0 c^3}, \] the Larmor formula: radiated power is proportional to the square of the charge and the square of its acceleration, and independent of the sphere radius.
The radiation reaction force is often treated as a small perturbation, valid only when \( \tau \dot{a} \ll a \). If the acceleration changes rapidly, this approximation breaks down, and the full relativistic equation of motion (Lorentz–Dirac) must be used, which introduces further complexities such as the Schott energy term. C
Worked example
Problem. How strong is radiation reaction for an electron? (a) Compute the electron's radiation time constant \(\tau = \mu_0 q^2/(6\pi m c)\). (b) An electron bound in an atom oscillates as \(x = x_0\cos\omega t\) with \(\omega = 3.0 \times 10^{15}\ \mathrm{rad\,s^{-1}}\) (visible light). Compare the amplitude of the radiation reaction force with that of the restoring force, and estimate how many oscillations it takes to radiate away most of its energy.
- (a) \(\tau = \dfrac{(1.257 \times 10^{-6})(1.60 \times 10^{-19})^2}{6\pi (9.11 \times 10^{-31})(3.00 \times 10^{8})} = \dfrac{3.22 \times 10^{-44}}{5.15 \times 10^{-21}} = 6.24 \times 10^{-24}\ \mathrm{s}\), the time light takes to cross about \(2 \times 10^{-15}\ \mathrm{m}\).
- (b) For \(x = x_0\cos\omega t\): \(a = -\omega^2 x\) and \(\dot a = \omega^3 x_0 \sin\omega t\). The reaction force \(m\tau\dot a\) has amplitude \(m\tau\omega^3 x_0\); the restoring force has amplitude \(m\omega^2 x_0\).
- Ratio: \(\omega\tau = 3.0 \times 10^{15} \times 6.24 \times 10^{-24} = 1.9 \times 10^{-8}\). The reaction force is a tiny perturbation, so treating it as weak damping is fully justified.
- Averaged over a cycle it acts like a damping term with energy decay rate \(\Gamma = \omega^2\tau = 5.6 \times 10^{7}\ \mathrm{s^{-1}}\), so the energy falls by \(1/e\) in \(1/\Gamma = 18\ \mathrm{ns}\), about \(1/(2\pi\omega\tau) \approx 8.5 \times 10^{6}\) oscillations.
Units check. \(\mu_0 q^2/(mc)\) has units \(\mathrm{N\,A^{-2}}\,\mathrm{A^2\,s^2}/(\mathrm{kg\,m\,s^{-1}}) = \mathrm{kg\,m\,s^{-2}\,s^2}/(\mathrm{kg\,m\,s^{-1}}) = \mathrm{s}\); \(\omega\tau\) is dimensionless and \(\omega^2\tau\) is in \(\mathrm{s^{-1}}\).
The nanosecond scale is the classical estimate of an excited atom's lifetime, the right order of magnitude for real atomic transitions.
Three quick tests
- Why does the radiation reaction force depend on the jerk \( \frac{d\vec{a}}{dt} \) rather than the acceleration \( \vec{a} \)?
Answer
It ensures that the work done by the force is proportional to the radiated power, which depends on \( a^2 \). The total derivative term integrates to zero over a cycle. - What is the physical significance of the time constant \( \tau \)?
Answer
It represents the characteristic timescale over which the radiation reaction effects become significant. It is extremely small for elementary particles, making radiation reaction a small perturbation in most laboratory scenarios. - Why are runaway solutions unphysical?
Answer
They imply that a free charge accelerates indefinitely without an external force, violating energy conservation and causality. They arise from the non-local nature of the radiation reaction force.
Problem set for PU-204 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes checked line by line by the editor; every number in the worked example was recomputed.