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Chapter 23: Problem 22

Cyclotron Motion with Radiation Reaction A non-relativistic particle with charge q performs circular cyclotron motion in a uniform magnetic field \(\mathbf{B}=B \hat{\mathbf{z}}\). Include the radiation reaction force \(m \tau_{0} \ddot{\mathbf{v}}\) in the equation of motion and solve it assuming that the motion remains approximately circular. Find the time constant for the decay of the particle velocity when the reaction is weak.

Short Answer

Expert verified
The time constant for decay is \( \tau = \frac{m^2}{q^2 B^2 \tau_0 v} \).

Step by step solution

01

Understand the Problem

We have a charged particle moving in a circular path due to the influence of a magnetic field. We need to include the effect of radiation reaction on the particle's motion and find how the particle's velocity decays over time.
02

Write the Equation of Motion

The usual equation of motion for a charge in a magnetic field is \(m \frac{d\mathbf{v}}{dt} = q(\mathbf{v} \times \mathbf{B})\). Including radiation reaction, the equation becomes \(m \frac{d\mathbf{v}}{dt} = q(\mathbf{v} \times \mathbf{B}) + m \tau_0 \ddot{\mathbf{v}}\), where \(\tau_0\) is a small constant related to the charge and the speed of light.
03

Simplify for Circular Motion

Assume the motion remains circular with constant angular frequency \(\omega\), so \(\mathbf{v} = v \hat{\mathbf{\theta}}\) where \(v\) is the speed. The magnetic force gives \(\omega = \frac{qB}{m}\). Now introduce the effect of radiation reaction: the change in velocity is a small decay term \(-\gamma \cdot \mathbf{v}\), assuming \(\gamma\) is small.
04

Substitute into the Equation

Substitute \(\dot{\mathbf{v}} = -\gamma \mathbf{v}\) and \(\ddot{\mathbf{v}} = \gamma^2 \mathbf{v}\) into our modified equation: \(m(-\gamma \mathbf{v}) = q(\mathbf{v} \times \mathbf{B}) + m \tau_0 (\gamma^2 \mathbf{v})\). Solve this expression for \(\gamma\).
05

Solve for the Time Constant

From the equation, the radiation reaction leads to a term \(m \gamma = m \tau_0 \cdot \gamma^2 + qBv\), simplifying under the assumption of weak reaction \((\gamma \ll \omega)\), giving \( \gamma = \frac{q^2 B^2 \tau_0}{m^2}v \). The time constant is \(\tau = \frac{1}{\gamma}\).
06

Calculate the Time Constant

Substitute the expression for \(\gamma\) to find the decay time constant \( \tau = \frac{m^2}{q^2 B^2 \tau_0 v} \). This gives the characteristic time over which the velocity decays to 1/e of its initial value due to radiation reaction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radiation Reaction
When a charged particle moves through a magnetic field, it emits electromagnetic radiation. This emitted radiation exerts a force back onto the particle itself. This is what we term as radiation reaction. Radiation reaction is a self-interaction force. It opposes the motion of the particle creating a drag-like effect, causing the particle to lose its kinetic energy over time.
  • The force due to radiation reaction is usually small and becomes significant only over long durations or high energies.
  • It's often described by a term involving the second time derivative of velocity, represented as \( m \tau_0 \ddot{\mathbf{v}} \).

Understanding radiation reaction is crucial for analyzing the long-term dynamics of charged particles in magnetic fields, especially in contexts such as synchrotrons and cyclotrons.
Equation of Motion
The equation of motion describes how a particle moves under the influence of forces. For a charge in a magnetic field, the Lorentz force is key. The Lorentz force is exerted by the magnetic and electric fields on a charged particle. It causes the particle to move in circular or helical paths.
  • The basic equation is \( m \frac{d\mathbf{v}}{dt} = q(\mathbf{v} \times \mathbf{B}) \), which captures this behavior.
  • Including the radiation reaction, the equation modifies to \( m \frac{d\mathbf{v}}{dt} = q(\mathbf{v} \times \mathbf{B}) + m \tau_0 \ddot{\mathbf{v}} \).

This addition accounts for the small decay in velocity due to radiation reaction, which is necessary for precision in scenarios like the cyclotron for particles at high velocities.
Magnetic Field
Magnetic fields exert forces on moving charges, causing them to move in specific paths. A uniform magnetic field has lines of force that are parallel and equally spaced, simplifying calculations.
  • In cyclotron motion, a charged particle enters a magnetic field and experiences a force perpendicular to both its velocity and the magnetic field, causing circular motion.
  • The angular frequency \( \omega = \frac{qB}{m} \) describes how quickly the particle orbits around the magnetic field lines.

Understanding the impact of magnetic fields on charged particles is essential in applications such as MRI machines and particle accelerators.
Non-relativistic Particle
Non-relativistic particles are those which move at speeds much lower than the speed of light. This simplifies many aspects of physics, allowing the use of classical mechanics instead of more complex relativistic physics.
  • For non-relativistic particles, calculations are simpler because we do not need to account for changes in mass with velocity.
  • In our context, this assumption means the particle鈥檚 mass remains constant, and relativistic effects like time dilation are negligible.

This assumption holds true for many laboratory-scale experiments and is a good approximation when dealing with cyclotron motion at typical speeds.

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Most popular questions from this chapter

Energy Lass and Electric Field Spectrum A charged particle produces an electric field E(r, \(t\) ) as it passes by an isolated atom. Model a bound electron in the atom as a damped harmonic oscillator with natural frequency os and damping constant \(\Gamma\). Assume that the electric field induces small- amplitude, nonrelativistic motion of the bound electron near the origin of coordinates. If \(\hat{\mathbf{E}}(\omega)\) is the Fourier transform of \(\mathbf{E}(\mathbf{r}=0, t)\), show that the energy transferred to the bound electron from the field is $$ \Delta E=\frac{e^{2}}{m \pi} \int_{0}^{\infty} d \omega|\hat{\mathbf{E}}(\omega)|^{2} \frac{\omega^{2} \Gamma}{\left(\omega^{2}-\omega_{0}^{2}\right)^{2}+\omega^{2} \Gamma^{2}} $$ What is \(\Delta E\) as \(\Gamma \rightarrow 0\) ?

The Covariant Li茅nard-Wiechert Field (a) Calculate \(F_{\mu \nu}=\partial_{\mu} A_{v}-\partial_{v} A_{\mu}\) from the Li茅nard-Wiechert potential \(A_{\mu}\). Express the result in terms of \(R_{\mu}\), the four-velocity \(U_{\mu}\), and the four-acceleration \(\partial U_{\mu} / \partial \tau\). Hint: It will be useful to evaluate \(\partial_{\mu}\left(R_{\sigma} R_{\sigma}\right)\) in order to show that \(\partial_{\mu} \tau=R_{\mu} /\left(R_{\sigma} U_{\sigma}\right)\). (b) Show that the splitting of \(F_{j e v}\) into a velocity part and an acceleration part is relativistically invariant. (c) Find \(\mathbf{E}\) and \(\mathbf{B}\) when \(\mathbf{v}=0\) (but \(\dot{\mathbf{v}} \neq 0\) ) by evaluating appropriate components of \(F_{\mu v}\) in the rest frame of the charge.

Angular Momentum Decay by Radiation Reaction Consider a classical atom where a particle with mass \(m\) and charge \(q\) orbits a fixed charge \(Z q\) non- relativistically. If \(L_{z}\) is the component of the angular momentum L that characterizes the orbit, show that the radiation reaction force \(m \tau_{0} \ddot{v}\) causes the energy and angular momentum of the "atom" to decay together such that $$ \frac{d L_{z}}{d E}=\frac{1}{2} \frac{L_{z}}{|E|} $$

The Direction of the Velocity Field Prove that the "velocity" part of the Lienard-Wiechert electric field points to the observer from the "anticipated position" of the moving point charge. The latter is the position the charge would have moved to if it retained the velocity \(\mathbf{v}_{\text {ret }}\) from \(t=t_{\text {ret }}\) to the present time of observation.

The Radiated Power Spectrum of a Linear Oscillator A point charge oscillates with the trajectory \(\mathbf{r}_{0}(t)=a \cos \left(\omega_{0} t\right) \hat{z}\). Find the angular distribution of power radiated into the \(m^{\text {th }}\) harmonic, \(d P_{w} / d \Omega\), during one period of the motion. Which harmonics dominate in the non-relativistic limit? Two Bessel function identities will be useful: $$ J_{m}(x)=\frac{i^{m}}{2 \pi} \int_{0}^{2 x} d \phi \exp [i(m \phi-x \cos \phi)] \quad \text { and } \quad J_{w+1}(x)+J_{m-1}(x)=(2 m / x) J_{w}(x) . $$
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