Question

A proton moving at speed \(v=1.00 \cdot 10^{6} \mathrm{~m} / \mathrm{s}\) enters a region in space where a magnetic field given by \(\vec{B}=\) \((-0.500 \mathrm{~T}) \hat{z}\) exists. The velocity vector of the proton is at an angle \(\theta=60.0^{\circ}\) with respect to the positive \(z\) -axis. a) Analyze the motion of the proton and describe its trajectory (in qualitative terms only). b) Calculate the radius, \(r\), of the trajectory projected onto a plane perpendicular to the magnetic field (in the \(x y\) -plane). c) Calculate the period, \(T,\) and frequency, \(f\), of the motion in that plane. d) Calculate the pitch of the motion (the distance traveled by the proton in the direction of the magnetic field in 1 period).

Short answer

Describe the trajectory of a proton moving with a velocity of \(1.00 \cdot 10^6 \mathrm{m/s}\) at an angle of \(60^{\circ}\) with the \(z\)-axis in a magnetic field of \(-0.500 \mathrm{T} \hat{z}\). Also, calculate its radius of the trajectory, time period, frequency, and pitch of the motion. The trajectory of the proton is in the shape of a helix. The radius of the trajectory is approximately \(5.77 \mathrm{mm}\), the time period is approximately \(6.58 \cdot 10^{-8} \mathrm{s}\), the frequency is approximately \(15.2 \mathrm{MHz}\), and the pitch of the motion is approximately \(3.29 \mathrm{mm}\).

Step-by-step solution

Calculate the magnetic force on the proton

The magnetic force on a charged particle can be found using the equation \(\vec{F} = q(\vec{v} \times \vec{B})\). In this case, the proton has charge \(q = 1.6 \cdot 10^{-19} \mathrm{C}\), velocity \(\vec{v} = 1.00 \cdot 10^{6} \mathrm{m/s}\) at an angle of \(60^{\circ}\) with the \(z\)-axis, and the magnetic field \(\vec{B} = -0.500 \mathrm{T} \hat{z}\).

Analyze and describe the motion trajectory

The proton's velocity has components in both the \(xy\)-plane and along the \(z\)-axis. In the \(xy\)-plane, the magnetic force is perpendicular to the velocity, causing the proton to move in a circular trajectory. Along the \(z\)-axis, there is no magnetic force, so the proton moves with constant velocity. As a result, the trajectory of the proton is in the shape of a helix.

Calculate the radius of the trajectory

To find the radius of the circular motion in the \(xy\)-plane, we can use the equation \(r = \frac{m v_\perp}{|q| B}\), where \(m\) is the mass of the proton, \(v_\perp\) is the component of the velocity perpendicular to the magnetic field, and \(B\) is the magnitude of the magnetic field. In this case, \(m = 1.67 \cdot 10^{-27} \mathrm{kg}\), and \(v_\perp = v\sin\theta\). Plugging in the values, we get: \(r = \frac{1.67 \cdot 10^{-27} \mathrm{kg} \cdot 10^6 \mathrm{m/s} \cdot \sin 60^{\circ}}{1.6 \cdot 10^{-19} \mathrm{C} \cdot 0.500 \mathrm{T}} \approx 5.77 \mathrm{mm}\).

Calculate the period and frequency of the motion

The time period of the circular motion in the \(xy\)-plane can be found using the equation \(T = \frac{2 \pi m}{|q| B}\), and the frequency is \(f = \frac{1}{T}\). Plugging in the values, we get: \(T = \frac{2 \pi \cdot 1.67 \cdot 10^{-27} \mathrm{kg}}{1.6 \cdot 10^{-19} \mathrm{C} \cdot 0.500 \mathrm{T}} \approx 6.58 \cdot 10^{-8} \mathrm{s}\), and \(f = \frac{1}{T} \approx 15.2 \mathrm{MHz}\).

Calculate the pitch of the motion

The pitch is the distance traveled by the proton along the \(z\)-axis in one period of the helical motion. It can be calculated as \(p = v_z T\), where \(v_z\) is the component of the velocity parallel to the magnetic field. Since the angle \(\theta\) is between the \(z\)-axis and the velocity vector, \(v_z = v\cos\theta\). Plugging in the values, we get: \(p = 10^6 \mathrm{m/s} \cdot \cos 60^{\circ} \cdot 6.58 \cdot 10^{-8} \mathrm{s} \approx 3.29 \mathrm{mm}\).

Key concepts

Magnetic Field Effects

The influence of a magnetic field on a charged particle is one of the fundamental concepts in physics, particularly in electromagnetism. When a charged particle, such as a proton or electron, enters a magnetic field, it experiences a force that affects its trajectory. This force does not work in the direction of the field, nor along the particle's initial direction of movement, but perpendicular to both the velocity of the particle and the magnetic field. This results in the charged particle altering its path, depending on the orientation of both its velocity and the magnetic field.

Quantifying Magnetic Field Effects

Mathematically, the magnetic field's effects on a charged particle can be quantified using the Lorentz force equation, which combines electrical and magnetic forces. For example, as seen in the exercise provided, the particle's velocity and the magnetic field vector are used to calculate the resulting force vector that determines the particle's new direction.

This relationship is essential in many technological applications, such as in the paths of charged particles in cyclotrons and other particle accelerators, the operation of mass spectrometers, and even in everyday devices like the cathode ray tubes in old televisions.

Helical Motion of Charged Particles

When a charged particle moves through a magnetic field at an angle, its motion becomes helical. This means it spirals around an invisible axis aligned with the magnetic field direction. In the exercise, the proton enters the field at a 60-degree angle to the magnetic field, resulting in a combination of circular motion in a plane perpendicular to the field and linear motion parallel to it.

Understanding Helical Motion

Helical motion occurs because the component of the particle's velocity perpendicular to the magnetic field results in circular motion (due to the magnetic force), while the component of velocity parallel to the field remains unaffected. Therefore, as the particle advances in the circular path, it simultaneously moves in the direction of the magnetic field, creating a helical trajectory. This knowledge is used in numerous scientific fields, including plasma physics and the study of helical structures in space plasmas.

Lorentz Force

The Lorentz force is a central principle in electromagnetism that describes the force experienced by a charged particle moving in an electromagnetic field. The magnetic component of the Lorentz force is given by the cross product of the velocity vector of the particle and the magnetic field vector, scaled by the charge of the particle.

Lorentz Force in Action

For a positively charged particle like the proton in the exercise, the Lorentz force equals the product of the charge, the speed of the particle, and the magnetic field magnitude, with the direction of the force perpendicular to both the velocity and the magnetic field. This force is responsible for the circular component of the helical motion discussed earlier. It is crucial to consider both the magnitude and direction of the force to understand the resulting motion of the charged particle, which is why the right-hand rule is often employed to determine the direction of the magnetic force.

Circular Motion in Magnetic Field

Circular motion occurs when a charged particle moves perpendicularly to a uniform magnetic field. This perpendicular motion means that the magnetic force acts as a centripetal force, constantly pulling the charged particle towards the center of a circular path, causing it to move in a circle at constant speed, as illustrated with the proton's path in the xy-plane in this exercise.

Characteristics of Circular Motion

Key characteristics of circular motion in a magnetic field include the radius of the circle and the period and frequency of the motion, which can be derived from the mass of the particle, its charge, the magnetic field strength, and the perpendicular component of the velocity. These factors give insight into how particles behave in magnetic confinement systems, like those used in fusion reactors or for trapping particles in magnetic bottles for experimentation.