Question

(i) A particle of rest mass \(m\) and charge \(q\) is injected at velocity \(\mathbf{u}\) into a constant pure magnetic field \(\mathbf{b}\) at right angles to the field lines. Use the Lorentz force law \((38.16)\) to establish that the particle will trace out a circle of radius \(m u \gamma(u) / q b\) with period \(2 \pi m \gamma(u) / q b\). [It was the y-factor in the period that necessitated the development of synchrotrons from cyclotrons, at whose energies the \(\gamma\) was still negligible.] (ii) If the particle is injected into the field with the same velocity but at an angle \(\theta \neq \pi / 2\) to the field lines, prove that the path is a helix, of smaller radius, but that the period for one complete cycle is the same as before.

Short answer

Answer: When the charged particle is moving perpendicular to the magnetic field lines, the radius of the motion is given by \(r = \frac{mu\gamma(u)}{qb}\), and the period is given by \(T = \frac{2\pi m\gamma(u)}{qb}\). When the angle between the initial velocity and magnetic field lines is not perpendicular, the particle traces a helix with a smaller radius \(r = \frac{mu\sin{\theta} \gamma(u)}{qb}\) but with the same period \(T = \frac{2\pi m\gamma(u)}{qb}\).

Step-by-step solution

Part (i): Charge moving perpendicular to magnetic field lines

1. Apply the Lorentz force law The Lorentz force acting on a charged particle moving in a magnetic field is given by \(\mathbf{F} = q(\mathbf{u} \times \mathbf{b})\). 2. Recognize the magnetic force is centripetal Since the particle's initial velocity is perpendicular to the magnetic field, the Lorentz force will always be perpendicular to the velocity. This means the force acts as a centripetal force, causing the particle to move in a circle. 3. Calculate the radius of the circle The centripetal force is given by \(\mathbf{F} = m\mathbf{a} = \frac{m\mathbf{u}^2}{r}\). Equate this with the Lorentz force: \(\frac{m\mathbf{u}^2}{r} = q(\mathbf{u} \times \mathbf{b})\). Use the fact that \(\mathbf{u} \times \mathbf{b} = ub\sin{\theta}\), and in this case, \(\theta = \pi/2\), so \(\sin{\theta} = 1\). Thus, we can write \(\frac{mu^2}{r} = qu- b\). Solving for the radius r, we get \(r = \frac{mu\gamma(u)}{qb}\). 4. Calculate the period of the motion To find the period, we can use the formula \(T = \frac{Distance}{Speed}\). Here, the distance is the circumference of the circle, \(2\pi r\), and the speed is constant and equal to u. Thus, \(T = \frac{2\pi r}{u}\). Substituting the expression for r, we get \(T = \frac{2\pi m\gamma(u)}{qb}\).

Part (ii): Charge moving with angle \(\theta\) to magnetic field lines

1. Decompose the initial velocity into parallel and perpendicular components We can decompose the initial velocity into components parallel (\(\mathbf{u}_{\parallel}\)) and perpendicular (\(\mathbf{u}_{\perp}\)) to the magnetic field. We have \(\mathbf{u}_{\parallel} = u\cos{\theta}\) and \(\mathbf{u}_{\perp} = u\sin{\theta}\). 2. Analyze the perpendicular motion The motion in the direction perpendicular to the magnetic field will be similar to the case in part (i). The magnetic force will be centripetal, and the radius of the helix will be given by \(r = \frac{mu_{\perp}\gamma(u)}{qb} = \frac{mu\sin{\theta} \gamma(u)}{qb}\). 3. Analyze the parallel motion The magnetic force acting in the direction parallel to the magnetic field is zero since the force is always perpendicular to the velocity. Therefore, the velocity in this direction will remain constant, and the helix will move with constant speed along the magnetic field lines. 4. Calculate the period of the helical motion Since there is no magnetic force acting in the parallel direction, the time it takes for one complete cycle of the helix will be the same as the time it takes for the perpendicular motion to complete one cycle. Thus, the period of the helical motion is the same as in part (i) and is given by \(T = \frac{2\pi m\gamma(u)}{qb}\). From this solution, we have shown that the particle traces out a circle with radius \(r = \frac{mu\gamma(u)}{qb}\) and period \(T = \frac{2\pi m\gamma(u)}{qb}\) when moving perpendicular to the magnetic field lines. When it starts with an angle \(\theta \neq \pi / 2\), it traces a helix with a smaller radius but with the same period.

Key concepts

Magnetic Field

A magnetic field is a force field created by moving electric charges or magnetic dipoles. It influences nearby moving charges and magnets. When a charged particle enters a magnetic field, it experiences a force called the Lorentz force, which acts perpendicular to both the direction of the field and the particle's velocity.

The strength and direction of a magnetic field are depicted by magnetic field lines. These lines run from the North to the South pole of a magnet. The density of these lines indicates the strength of the field.

A uniform magnetic field, like the one described in many physics problems, has parallel lines of constant strength and direction. In this context, the magnetic field influences charged particles by altering their paths without changing their speed.

Centripetal Force

Centripetal force is the force required to keep an object moving in a curved path and is directed towards the center around which the object is moving. In the context of a charged particle moving in a magnetic field, the Lorentz force provides this centripetal force.

Consider a particle moving perpendicular to a magnetic field:

• The magnetic force acts as a centripetal force, causing the particle to move in a circular path.
• The size of the circle is determined by factors like the particle's mass, charge, velocity, and the strength of the magnetic field.

The centripetal force equation can be expressed as: \[ F = \frac{mv^2}{r} = qvB \] where \(m\) is mass, \(v\) is velocity, \(r\) is radius, \(q\) is charge, and \(B\) is the magnetic field strength. This relationship helps us understand why a charged particle travels in circular paths under the influence of a magnetic field.

Cyclotron

A cyclotron is a type of particle accelerator that uses magnetic fields to spiral particles in a circular path, allowing them to gain energy.

Here's a simple breakdown of how it works:

• Charged particles are injected into a cyclotron chamber.
• A magnetic field forces them into circular paths.
• Electric fields applied at specific times accelerate the particles, increasing their speed with each pass.

When particles gain enough energy, they exit the cyclotron to be used in various experiments or applications, such as medical treatments.

Cyclotrons can only accelerate particles to non-relativistic speeds, as the approximation becomes less accurate as particles near the speed of light. At higher energies, where the relativistic effects are significant, a modification of the cyclotron, known as the synchrotron, is employed.

Synchrotron

A synchrotron is an advanced type of cyclotron designed to accelerate particles to much higher speeds. Unlike cyclotrons, synchrotrons can handle the relativistic effects that become significant at high velocities.

Key features of a synchrotron include:

• Variable magnetic fields: Adjusting the magnetic field synchronously with the particle's energy ensures they continue to follow a circular path.
• RF cavities: Used to accelerate particles progressively as they travel around the synchrotron ring.

Synchrotrons have a crucial role in advanced scientific research, producing high-energy beams for probing matter at the atomic scale.

The transition from cyclotrons to synchrotrons was necessary to overcome energy limitations and explore new frontiers in particle physics and materials science.