Question

Cosmic rays from space that strike Earth contain some charged particles with energies billions of times higher than any that can be produced in the biggest accelerator. One model that was proposed to account for these particles is shown schematically in the figure. Two very strong sources of magnetic fields move toward each other and repeatedly reflect the charged particles trapped between them. (These magnetic field sources can be approximated as infinitely heavy walls from which charged particles get reflected elastically.) The high- energy particles that strike the Earth would have been reflected a large number of times to attain the observed energies. An analogous case with only a few reflections demonstrates this effect. Suppose a particle has an initial velocity of \(-2.21 \mathrm{~km} / \mathrm{s}\) (moving in the negative \(x\) -direction, to the left), the left wall moves with a velocity of \(1.01 \mathrm{~km} / \mathrm{s}\) to the right, and the right wall moves with a velocity of \(2.51 \mathrm{~km} / \mathrm{s}\) to the left. What is the velocity of the particle after six collisions with the left wall and five collisions with the right wall?

Short answer

Answer: The final velocity of the charged particle is -2.81 km/s.

Step-by-step solution

Establish the initial conditions

Initially, the particle has a velocity of \(-2.21 \; km/s\), the left wall has a velocity of \(1.01 \; km/s\), and the right wall has a velocity of \(-2.51 \; km/s\).

Calculate the relative velocities

Before each collision, we need to calculate the relative velocity of the particle with respect to each wall. The relative velocity of the particle with respect to the left wall is \(v_{p,left} = -2.21\; km/s - 1.01\; km/s = -3.22 \; km/s\). Similarly, the relative velocity of the particle with respect to the right wall is \(v_{p,right} = -2.21\; km/s - (-2.51\; km/s) = 0.3\; km/s\).

Reflect the particle after a collision

After each collision, the relative velocity of the particle with respect to the wall is reversed. Hence, after a collision with the left wall, the relative velocity of the particle with respect to the left wall is \(-v_{p,left} = 3.22 \; km/s\). Similarly, after a collision with the right wall, the relative velocity of the particle with respect to the right wall is \(-v_{p,right} = -0.3\; km/s\).

Repeat the process for 6 collisions with the left wall and 5 collisions with the right wall

After each collision, we need to add up the relative velocity of the particle with respect to the wall and the velocity of the wall itself to get the velocity of the particle after the collision. We'll do this repeatedly for 6 collisions with the left wall and 5 collisions with the right wall: For 6 collisions with the left wall: \(v_{1} = 3.22 \; km/s + 1.01 \; km/s = 4.23 \; km/s\) \(v_{2} = -3.22 \; km/s + 1.01 \; km/s = -2.21 \; km/s\) \(v_{3} = 3.22 \; km/s + 1.01 \; km/s = 4.23 \; km/s\) \(v_{4} = -3.22 \; km/s + 1.01 \; km/s = -2.21 \; km/s\) \(v_{5} = 3.22 \; km/s + 1.01 \; km/s = 4.23 \; km/s\) \(v_{6} = -3.22 \; km/s + 1.01 \; km/s = -2.21 \; km/s\) For 5 collisions with the right wall: \(v_{1} = -0.3\; km/s + (-2.51 \; km/s) = -2.81 \; km/s\) \(v_{2} = 0.3\; km/s + (-2.51 \; km/s) = -2.21 \; km/s\) \(v_{3} = -0.3\; km/s + (-2.51 \; km/s) = -2.81 \; km/s\) \(v_{4} = 0.3\; km/s + (-2.51 \; km/s) = -2.21 \; km/s\) \(v_{5} = -0.3\; km/s + (-2.51 \; km/s) = -2.81 \; km/s\)

Determine the final velocity of the particle

After six collisions with the left wall, the particle has a velocity of \(-2.21 \; km/s\). After five collisions with the right wall, the particle has a velocity of \(-2.81 \; km/s\). Therefore, the final velocity of the particle is \(-2.81\; km/s\).

Key concepts

Charged Particles

Charged particles, like the ones found in cosmic rays, are particles that carry an electrical charge. These particles can be electrons or protons, and their charge makes them susceptible to electromagnetic forces. Charged particles are accelerated by electric fields and deflected by magnetic fields. When they are caught between two moving magnetic fields, their behavior becomes particularly interesting.
Understanding how these particles interact with fields is crucial because it helps explain phenomena from everyday technology to cosmic events. Charged particles are not only theoretical constructs but are involved in practical applications like cathode-ray tubes and particle accelerators.

Magnetic Fields

Magnetic fields are invisible forces generated by magnetic objects, such as magnetic field sources in the exercise. These fields exert a force on charged particles, causing them to move in circular or spiral paths.
Magnetic fields are typically described by field lines that illustrate the direction and strength of the magnetic force. They impact charged particles by changing their direction, not their speed, which can explain why particles reflect off the fields as if hitting a wall. This effect is paramount in many physics applications, including keeping particles within paths in accelerators and protecting Earth from cosmic radiation.

Velocity Calculation

Velocity is a vector quantity that has both magnitude and direction. When calculating velocities, especially in elastic collisions, knowing the relative velocity between objects is essential.
In this exercise, the velocity of the particle changes due to the movement of magnetic fields, likened to walls. The steps involve calculating the velocity of the particle relative to each wall before and after collisions.
Use the formula:

• Relative velocity before collision = velocity of particle - velocity of wall
• Relative velocity after collision = - (relative velocity before collision)

This equation helps in determining how velocities change due to interactions between particles and walls, verifying the principle of conservation of momentum in elasticity.

Elastic Collisions

An elastic collision is a type of collision where the total kinetic energy of the system is conserved. This means that the sum of the kinetic energy before the collision is equal to the sum after the collision.
In the given problem, particles collide elastically with moving magnetic field sources.
Key points of elastic collisions include:

• Conservation of kinetic energy
• Absence of permanent deformation or generation of heat
• Reversal of relative velocity across colliding surfaces

For instance, in elastic collisions between the charged particle and the magnetic walls, the speed doesn't change although kinetic energy and momentum are transferred. This ability to conserve energy allows cosmic particles to gain significant energy after numerous reflections, explaining their high energies when striking the Earth.