Problem 13 - Physics for Scientists and Engineers with Modern Physics

Question

(II) Show that the energy of a particle (charge e) in a synchrotron, in the relativistic limit \((v \approx c),\) is given by \(E(\) in 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

The energy of the particle is \(E = B r c\).

Step 1: Understand the Synchrotron Motion

In a synchrotron, a charged particle moves in a circular path due to the presence of a magnetic field. In the relativistic limit, where the particle velocity \(v\) is nearly equal to the speed of light \(c\), the centripetal force required to keep the particle in circular motion is provided by the magnetic force.

Step 2: Write the Force Equation

The magnetic force \(F_B\) acting on a particle with charge \(e\) moving with velocity \(v\) in a magnetic field \(B\) is given by \(F_B = e v B\).For circular motion, this force provides the necessary centripetal force \(F_c\), which is \(F_c = \frac{m v^2}{r}\). Here \(m\) is the relativistic mass of the particle. In equilibrium, \(F_B = F_c\).

Step 3: Substitute Relativistic Mass

In the relativistic regime, the relativistic mass \(m\) is defined as \(m = \gamma m_0\), where \(m_0\) is the rest mass and \(\gamma\) is the Lorentz factor \(\gamma = \frac{1}{\sqrt{1 - (v^2/c^2)}}\). However, as \(v \approx c\), \(\gamma \approx \frac{1}{\sqrt{1-(c^2/c^2)}}\) approaches infinity, simplifying consideration of terms.

Step 4: Define Relativistic Energy

The total relativistic energy \(E\) of the particle is given by \(E = \gamma m_0 c^2\). Due to high value of \(\gamma\) as \(v \approx c\), \(E\) becomes approximately equal to \(\gamma m_0 c^2\).

Step 5: Solve for Energy in a Synchrotron

From step 2, we have \(e v B = \frac{m v^2}{r}\). For relativistic speeds, \(v \approx c\), hence we replace \(v\) with \(c\): \[ e c B = \frac{E}{c r} \]Therefore, rearranging gives:\[ E = e B r c \]This shows that the energy \(E\) in electron volts is indeed proportional to \(B r c\), assuming charge \(e\) is expressed in terms of elementary charge units.

Key Concepts

Relativistic mass

When particles move close to the speed of light, their mass isn't the same as when they are at rest. This concept is known as relativistic mass. As a particle speeds up, it appears more massive. This is due to the effects of relativity described by Einstein. In simple terms, the relativistic mass formula is \[ m = \gamma m_0 \]where:

• \( m \) is the relativistic mass.
• \( m_0 \) is the rest mass (mass of the particle when it's not moving).
• \( \gamma \) is the Lorentz factor, which we will discuss in more detail soon.

As a particle's speed increases, its relativistic mass increases as well due to the rise in the Lorentz factor. This concept is vital in understanding how particles behave in high-speed environments like synchrotrons.

Magnetic field in synchrotron

A synchrotron utilizes a magnetic field to control the motion of charged particles traveling at high velocities. In these devices, the magnetic field has an essential function. It bends the paths of fast-moving particles, forcing them into circular trajectories.The calculation of the magnetic force acting on a particle in a synchrotron is based on the equation:\[ F_B = e v B \]where:

• \( e \) is the charge of the particle.
• \( v \) is the velocity of the particle.
• \( B \) is the magnetic field strength.

In the relativistic context, when the particle's velocity \( v \) approaches the speed of light \( c \), the magnetic force equals the centripetal force needed to maintain the circular motion. This relationship is crucial in maintaining a stable orbit for particles in synchrotrons.

Centripetal force in circular motion

Centripetal force is necessary to keep an object moving in a circle. When a particle travels in a circular path, a force must act towards the center of the circle. Without this force, the object would continue moving in a straight line due to inertia.In synchrotrons, the centripetal force is matched by the magnetic force. The equation for centripetal force is:\[ F_c = \frac{m v^2}{r} \]where:

• \( m \) is the mass of the particles.
• \( v \) is the velocity.
• \( r \) is the radius of the circular path.

For particles at relativistic speeds, this relationship becomes\[ e v B = \frac{m v^2}{r} \]since the magnetic force \( F_B \) provides the necessary centripetal force \( F_c \). This balance allows synchrotrons to keep particles in stable orbits.

Lorentz factor

The Lorentz factor is crucial when dealing with particles moving at speeds close to the speed of light. Denoted as \( \gamma \), it accounts for the effects of special relativity on time, length, and relativistic mass.The Lorentz factor is calculated as:\[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]where:

• \( v \) is the velocity of the particle.
• \( c \) is the speed of light.

When \( v \) is much less than \( c \), \( \gamma \approx 1 \). However, when \( v \) nears \( c \), \( \gamma \) becomes significantly larger, influencing the relativistic mass. This factor makes sure that the laws of physics adjust accurately for fast moving particles, such as those in synchrotrons.