Problem 10 - Classical Mechanics

Question

A beam of particles of charge \(q\) and velocity \(v\) is emitted from a point source, roughly parallel with a magnetic field \(\boldsymbol{B}\), but with a small angular dispersion. Show that the effect of the field is to focus the beam to a point at a distance \(z=2 \pi m v /|q| B\) from the source. Calculate the focal distance for electrons of kinetic energy \(500 \mathrm{eV}\) in a magnetic field of \(0.01 \mathrm{~T}\). (Charge on electron \(=-1.6 \times 10^{-19} \mathrm{C}\), mass \(=9.1 \times 10^{-31} \mathrm{~kg}\), \(\left.1 \mathrm{eV}=1.6 \times 10^{-19} \mathrm{~J} .\right)\)

Step-by-Step Solution

Answer

Question: Calculate the focal distance of a beam of charged particles emitted from a point source, moving with an electron kinetic energy of 500 eV, in the presence of a magnetic field of 0.01 T. Answer: The focal distance for the given electron is approximately 2.15 x 10^{-2} meters.

Step 1: 1. Determine the Force on the Particles

For a charged particle with charge \(q\) and velocity \(\boldsymbol{v}\) in a magnetic field \(\boldsymbol{B}\), the force exerted on the particle is given by the Lorentz force \(\boldsymbol{F}_{B}=q \boldsymbol{v} \times \boldsymbol{B}\).

Step 2: 2. Find the Equation of Motion for the Charged Particles

The equation of motion for a particle with mass \(m\) under the action of the Lorentz force is given by Newton's second law: \(\boldsymbol{F}_{B} = m \boldsymbol{a}\), where \(\boldsymbol{a}\) is the acceleration of the particle. Substituting the Lorentz force, we get: \(q \boldsymbol{v} \times \boldsymbol{B} = m \boldsymbol{a}\).

Step 3: 3. Find the Relation Between \(z\) and other Variables

From the previous equation, we can find that the acceleration is perpendicular to both \(\boldsymbol{v}\) and \(\boldsymbol{B}\). The centripetal force exerted by the magnetic field on the particle, causing it to move in a circular path, is given by \(F_{c}=m \omega^{2} R\), where \(R\) is the radius of the circular path and \(\omega\) is the angular frequency. Since \(F_{c}=|F_{B}|\), and \(|F_{B}|=qvB\), we can write: \(m \omega^{2} R=|q| v B\). We also have \(\omega=\frac{2\pi v}{z}\) due to the focusing of particles at distance \(z\). Substituting this into the previous equation, we can find the relation between \(z\) and the other variables: \(z=2\pi \frac{mv}{|q|B}\).

Step 4: 4. Calculate the Focal Distance

Given the kinetic energy \(E_{k} = 500\) eV, we can find the velocity of the electrons by using the relation \(E_{k} = \frac{1}{2} m v^{2}\). Plugging in the values for mass, charge, and magnetic field, we find: \(v=\sqrt{\frac{2 \times 500 \times 1.6 \times 10^{-19}}{9.1 \times 10^{-31}}} \text{m/s} \approx 1.32 \times 10^{6} \text{m/s}\). Substituting the values in the expression for \(z\), we find the focal distance: \(z = \frac{2\pi \times 9.1 \times 10^{-31} \times 1.32 \times 10^{6}}{1.6 \times 10^{-19}\times 0.01} \approx 2.15 \times 10^{-2} \text{m}\).

Key Concepts

Lorentz Force

Imagine holding a magnet near a moving charged particle, such as an electron. The particle will curve away from its straight path. This is due to a fundamental force known as the Lorentz force. It occurs when a charged particle, with charge q, moves with velocity v through a magnetic field B. The resulting force, mathematically represented as F_B = q(v × B), is perpendicular to both the velocity of the particle and the direction of the magnetic field.

This force is central in explaining how particle beams, like the one described in our exercise, are able to bend and focus. The ability to calculate and predict the impact of the Lorentz force is crucial in designing equipment for scientific experiments, such as cyclotrons and spectrometers, which rely on the manipulation of particle trajectories using magnetic fields.

Equation of Motion

The equation of motion describes how an object moves through space over time. In the context of particles moving through a magnetic field, we use Newton's second law to relate the Lorentz force to the acceleration of the particle: F_B = m * a, where m is the mass of the particle and a is its acceleration. This equation allows us to predict the future position and velocity of a particle under the influence of a force.

In our exercise, once we substitute the Lorentz force into the equation of motion, we can see why the particle undergoes circular motion. The acceleration, being perpendicular to both velocity and magnetic field, points towards the centre of the particle's curved path, maintaining its circular motion. This insight is essential for calculating the path of a particle in various applications, including particle accelerators where precise control of particle motion is needed.

Centripetal Force

When a particle moves in a circle, it experiences an inward force that keeps it on its circular path. This inward force is called the centripetal force, which for a particle in uniform circular motion is given by F_c = mω²R, where R is the radius of the circular path, and ω is the angular frequency of the particle.

In our exercise, the centripetal force is provided by the magnetic field exerting the Lorentz force. Every particle within the beam, no matter its angle of dispersion, experiences this centripetal force, leading to the focus of the beam at a particular point. Understanding the relationship between centripetal force and the magnetic forces at play is vital for tasks like bending particle beams in a controlled manner, which is a common procedure in medical imaging technologies.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. It is given by the equation E_k = 1/2 m v², where m is the mass and v is the velocity of the object. The concept of kinetic energy helps us understand the work done on a particle and the energy it carries as it moves.

In the context of our exercise, when an electron beam with a known kinetic energy is emitted into a magnetic field, we can determine the velocity of the particles using the kinetic energy equation. This velocity is then used to calculate the focusing distance of the beam, which shows how energy translates into motion and, under the influence of a magnetic field, becomes a precise focal point. Grasping kinetic energy concepts is crucial for any application that involves motion, such as the transport of electrons in electrical circuits or the calculation of the trajectories in astrophysical models.