Question

A proton with an initial velocity given by \((1.00 \hat{x}+2.00 \hat{y}+3.00 \hat{z})\) \(\left(10^{5} \mathrm{~m} / \mathrm{s}\right)\) enters a magnetic field given by \((0.500 \mathrm{~T}) \hat{z}\). Describe the motion of the proton.

Short answer

**Short Answer:** The proton will move in an elliptical helix path.

Step-by-step solution

Write down the formula for the Lorentz force

The Lorentz force acting on a charged particle with charge q in a magnetic field B is given by F = q(v x B), where v is the velocity of the particle and x represents the cross product.

Plug in the values for the proton's initial velocity and magnetic field

We are given the initial velocity of the proton as \((1.00 \hat{x}+2.00 \hat{y}+3.00 \hat{z})\left( 10^5 \mathrm{~m/s} \right)\) and the magnetic field as \((0.500 \mathrm{~T}) \hat{z}\). The charge of the proton is \(1.6\times10^{-19} \mathrm{C}\). Now, we can calculate the cross product of the velocity and magnetic field vectors.

Calculate the cross product of the initial velocity and magnetic field vectors

The cross product is given by: $v \times B = \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \\ 1.00\times10^{5} & 2.00\times10^{5} & 3.00\times10^{5} \\ 0 & 0 & 0.500 \end{vmatrix}$ Expanding this determinant, we get: \(v \times B = (0.500\times2.00\times10^5)\hat{x} - (0.500\times1.00\times10^5)\hat{y} = (1.00\times10^5 \hat{x} - 0.500\times10^5 \hat{y})\text{T m/s}\)

Calculate the Lorentz force

Now, we can calculate the Lorentz force using the formula F = q(v x B): \(F = q(v \times B) = (1.6\times10^{-19} \mathrm{C}) \times (1.00\times10^5 \hat{x} - 0.500\times10^5 \hat{y})\text{T m/s}\) \(F = (1.6\times10^{-14}\hat{x} - 0.8\times10^{-14}\hat{y})\text{N}\)

Describe the motion of the proton

The Lorentz force on the proton is perpendicular to both the velocity and magnetic field vectors. This force will cause the proton to move in a circular path in the xy-plane, with the z-component of the velocity remaining constant. The direction of the centripetal force will be along the x-axis, causing the radius of the circular path to be greater in the x-direction compared to the y-direction. Thus, the proton will move in an elliptical helix path in the presence of the given magnetic field.

Key concepts

Motion of Charged Particles

Understanding the behavior of charged particles, such as protons and electrons, when they enter a magnetic field is a fundamental concept in physics. Charged particles experience a force when they move within a magnetic field, which affects their trajectories. In our exercise, a proton with velocity enters a uniform magnetic field and this interaction is governed by the Lorentz force.

The motion of the proton can be predicted using the concept of the Lorentz force, which acts perpendicular to the velocity of the proton and the magnetic field. Since this force is always perpendicular to the direction of motion, it doesn't do work on the proton and therefore doesn't change its kinetic energy; however, it can change the direction of the proton's velocity. In a uniform magnetic field, a charged particle moves in a circular orbit in the plane perpendicular to the field lines. The radius of this orbit depends on the mass, speed, and charge of the particle, as well as the strength of the magnetic field.

However, if the velocity of the particle has a component along the direction of the magnetic field, this part of the velocity is unaffected, and the particle moves in a helical (spiral) path. In the exercise, the proton moves in an elliptical helix, combining circular motion in the xy-plane with linear motion along the z-axis.

Magnetic Fields in Physics

Magnetic fields are regions of space around magnetic materials or moving electric charges within which the force of magnetism acts. The magnetic field vector, often denoted as B, represents both the direction and the magnitude of the field. In physics, the interactions between charged particles and magnetic fields are described by electromagnetism, which is one of the four fundamental forces.

Magnetic fields can be visualized through the use of field lines, which show the direction of the magnetic force. These lines emerge from the north pole of a magnet and enter at the south pole. Charged particles are influenced by magnetic fields only when they possess a velocity component that is perpendicular to the direction of the magnetic field. When a charged particle moves parallel to the magnetic field lines, it doesn't experience any force.

In the given exercise, we have a magnetic field aligned along the z-axis. This setup creates an effect where the proton, possessing initial velocity components in the x, y, and z directions, experiences a force that acts only within the plane perpendicular to the magnetic field, leading to the described helical motion.

Cross Product of Vectors

The cross product, also known as the vector product, is an operation on two vectors in three-dimensional space, resulting in a new vector that is perpendicular to both of the original vectors. This tool is essential for understanding many physical concepts, including the Lorentz force, which is crucial for predicting the motion of charged particles in a magnetic field.

The cross product of vectors A and B, denoted as A x B, can be calculated using the determinant of a matrix composed of the standard unit vectors along the coordinate axes and the components of the two vectors. The order of the vectors in the cross product matters, as A x B is the negative of B x A.

In this specific problem, the cross product calculation effectively determines the direction and magnitude of the Lorentz force that acts on the proton. Since the velocity of the proton has components in all three axes (x, y, and z), and the magnetic field is directed along the z-axis, the resulting force from the cross product has components in the x and y directions, with no component along the magnetic field. This perpendicular nature of the force causes the proton to move in a circular path within the xy-plane while maintaining a constant velocity in the z direction, thereby creating the elliptical helical motion.