Problem 13 - Physics for Scientists and Engineers with Modern Physics

Question

Show that the energy of a particle (charge \(e\) ) in a synchrotron, in the relativistic limit \((v \approx c),\) is given by \(E(\) in \(\mathrm{eV})=B r c,\) where \(B\) is the magnetic field and \(r\) is the radius of the orbit (SI units).

Step-by-Step Solution

Answer

In the relativistic limit, the energy of a particle in a synchrotron, in eV, is given by the expression \(E = Brc\).

Step 1: Relate Energy and Momentum

In the relativistic framework, the energy of a particle is given by the expression \(E^2 = p^2c^2 + m^2c^4\), where \(E\) is the energy, \(p\) is the momentum, and \(m\) is the rest mass. For a particle moving at speeds close to the speed of light \( (v \approx c)\), the rest mass energy \( m^2c^4 \) is negligible compared to the kinetic energy, so we consider \(E \approx pc\).

Step 2: Express Momentum in Terms of Synchrotron Parameters

For a particle with charge \(e\) moving in a magnetic field \(B\) along a circular path of radius \(r\), the centripetal force is provided by the magnetic Lorentz force: \(evB = \frac{mv^2}{r}\). In the relativistic regime, \( p = \gamma mv \) is used, where \( \gamma \) is the Lorentz factor. However, we focus on the classical formula as momentum for a circular motion: \(p = eBr\).

Step 3: Substitute Momentum in Energy Equation

Substitute the expression for momentum \(p = eBr\) into the energy equation \(E \approx pc\) from Step 1. This gives us \(E \approx (eBr)c\), which simplifies further to \(E = eBrc\).

Step 4: Convert Energy Units

The result \(E = eBrc\) is in joules since the equation uses SI units. To convert it to electronvolts (eV), we note that 1 eV = \(1.60219 \times 10^{-19}\) J. However, as per the problem statement, the constants are aligned to give units directly in eV when \(B\) is in Tesla, \(r\) in meters, and \(c\) in meters per second. Thus no further conversion is needed.

Key Concepts

Relativistic Physics

In the realm of relativistic physics, as velocities approach the speed of light (\( c \), which is approximately 299,792,458 meters per second), the behavior of particles deviates from classical predictions. Traditional equations of motion no longer suffice, and the effects of relativity must be considered to accurately describe a particle's energy and momentum.

The energy relation in relativistic physics is described by the equation \( E^2 = p^2c^2 + m^2c^4 \), where \( E \) is the total energy, \( p \) represents the relativistic momentum, and \( m \) is the rest mass of the particle.

However, in scenarios where a particle's velocity is very close to \( c \), the rest mass energy component becomes negligible compared to the kinetic energy component. Thus, the energy is often approximated as \( E \approx pc \), simplifying calculations and making it feasible to analyze high-speed particles in synchrotron setups.

Magnetic Lorentz Force

The magnetic Lorentz force is a fundamental concept explaining how particles behave in magnetic fields. For a charged particle with charge \( e \) moving through a magnetic field \( B \) at velocity \( v \), the magnetic Lorentz force can be defined as:

• \( F = evB \) when the particle's velocity is perpendicular to the magnetic field.

The force causes the particle to move in a circular path with a radius \( r \). This is because the Lorentz force acts as the centripetal force required for circular motion.

In a synchrotron, a ring of magnetic fields is used to maintain this circular motion by continuously applying the Lorentz force. From the classical reference, we know centripetal force can be equated to the magnetic Lorentz force as \( \frac{mv^2}{r} = evB \). In the relativistic context, though classically given by \( p = eBr \), it is more accurately described by \( p = \gamma mv \) due to relativistic momentum considerations. Here, \( \gamma \) is the Lorentz factor, accounting for the time dilation and length contraction effects at near-light speeds.

Kinetic Energy in Relativistic Motion

Kinetic energy in relativistic motion involves more complex calculations than classical mechanics. In synchrotrons, where particles move at speeds close to light, this concept becomes particularly relevant.

A particle’s kinetic energy increases disproportionately as its speed approaches \( c \), unlike in classical physics where kinetic energy is simply \( \frac{1}{2}mv^2 \). In relativistic terms, kinetic energy becomes significant in relation to \( E = pc \), especially considering the previously mentioned approximation when rest mass energy is negligible.

In the setup of a synchrotron, this kinetic energy is directly linked to the magnetic Lorentz force and play a significant role in determining the particle’s energy \( E \) as given by the convenient form \( E = eBrc \). Using this form captures the essence of the relationship between the synchrotron's magnetic field, particle path, and energy output. This makes it easier to compute and understand how relativistic speeds affect kinetic energy by binding movement, force, and energy into a streamlined equation.