Question

A charged particle of mass \(m\) and positive charge \(q\) moves in uniform electric and magnetic fields, \(\mathbf{E}\) and \(\mathbf{B}\), both pointing in the \(z\) direction. The net force on the particle is \(\mathbf{F}=q(\mathbf{E}+\mathbf{v} \times \mathbf{B})\). Write down the equation of motion for the particle and resolve it into its three components. Solve the equations and describe the particle's motion.

Short answer

The particle follows a helical path, spiraling around the z-axis with linear acceleration due to the electric field.

Step-by-step solution

Write Down the Force Equation

The force on the particle is given by the Lorentz force: \( \mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \). Where \( \mathbf{v} \) is the velocity of the particle, \( \mathbf{E} = E \hat{k} \) (electric field in the z-direction), and \( \mathbf{B} = B \hat{k} \) (magnetic field in the z-direction).

Express the Cross Product

Cross product \( \mathbf{v} \times \mathbf{B} \) is expanded to: \( \begin{pmatrix} v_x \ v_y \ v_z \end{pmatrix} \times \begin{pmatrix} 0 \ 0 \ B \end{pmatrix} = \begin{pmatrix} v_y B \ -v_x B \ 0 \end{pmatrix}.\)

Breakdown into Component Equations

Substitute the cross product result and resolve the force equation into components:- In the x-direction: \( F_x = q(v_y B) \)- In the y-direction: \( F_y = -q(v_x B) \)- In the z-direction: \( F_z = q(E) \)

Convert to Equations of Motion

The equations of motion using Newton's second law \( F = m \frac{d^2 r}{dt^2} \) for each component:1. \( m \frac{d^2 x}{dt^2} = qv_y B \)2. \( m \frac{d^2 y}{dt^2} = -qv_x B \)3. \( m \frac{d^2 z}{dt^2} = qE \)

Solve the Equations of Motion

1. Z-direction: Integrating \( m \frac{d^2 z}{dt^2} = qE \) results in uniform acceleration \( z(t) = \frac{qE}{m} \frac{t^2}{2} + v_{z0} t + z_0 \).2. X and Y-directions are coupled: - Substitute for cyclotron frequency \( \omega = \frac{qB}{m} \), leading to harmonic motion: - \( \frac{d^2 x}{dt^2} = \omega \frac{dy}{dt} \) - \( \frac{d^2 y}{dt^2} = -\omega \frac{dx}{dt} \) - These result in circular motion: solutions are \( x(t) = R \cos(\omega t + \phi_x) \) and \( y(t) = R \sin(\omega t + \phi_y) \).

Describe the Motion

The motion decomposes into two parts: 1. Helical motion around the z-axis with radius \( R \) due to the magnetic component causing circular motion in the xy-plane.2. Accelerated linear motion in the z-direction due to the electric field.

Key concepts

Charged Particle Motion

When a charged particle, such as an electron or proton, enters a region with external fields, its path can change dramatically depending on the field types and the particle’s initial velocity. Understanding charged particle motion is key in fields like electromagnetism and plasma physics.
This motion is primarily governed by the Lorentz force, which dictates that the combined effects of electric and magnetic fields will influence how the particle accelerates. This means once you know the charge, the initial velocity, and the configuration of the fields, you can predict the particle’s trajectory.
In our example, the electric and magnetic fields are uniform and directed along the same axis (z-direction). This simplification lets us predict two main types of motion: circular motion caused by the magnetic field and linear motion influenced by the electric field.

Electric and Magnetic Fields

These fields play a crucial role in determining the behavior of charged particles. An electric field (\(\mathbf{E}\)) exerts a force straight along the field lines, impacting the particle's motion along that direction. The magnetic field (\(\mathbf{B}\)), by contrast, influences motion perpendicular to itself and the velocity—which results in circular dynamics.
The fields in the given problem are both aligned in the z-direction. The electric field \(\mathbf{E} = E \hat{k}\), where \(E\) is the field strength, accelerates the particle along the z-axis. The magnetic field \(\mathbf{B} = B \hat{k}\) causes motion in the xy-plane via the Lorentz force expression \(\mathbf{v} \times \mathbf{B}\).
This means electric fields typically drive acceleration and can give particles kinetic energy along their direction, while magnetic fields are known for inducing rotational or circular motion.

Equations of Motion

To precisely determine the particle's path, we describe its motion mathematically using equations based on Newton's Second Law. We resolve these equations into three components—x, y, and z-axes—correlating to the directions relative to the fields.
The generalized force equation \(\mathbf{F} = m \frac{d^2 \mathbf{r}}{dt^2}\), when combined with the Lorentz force \(q(\mathbf{E} + \mathbf{v} \times \mathbf{B})\), becomes the basis for our component equations:

• In the x-direction: \(m \frac{d^2 x}{dt^2} = q v_y B\)
• In the y-direction: \(m \frac{d^2 y}{dt^2} = -q v_x B\)
• In the z-direction: \(m \frac{d^2 z}{dt^2} = q E\)

Solving these uncovers the particle's motion: a linear path along the z-axis and circular motion in the xy-plane due to the different influences of the fields.

Cyclotron Frequency

A particularly interesting result from charged particle dynamics in magnetic fields is the cyclotron frequency, \(\omega = \frac{qB}{m}\). This represents the rate at which a particle circles in the plane perpendicular to a magnetic field.
This frequency is a constant for given values of charge, magnetic field strength, and mass, meaning it depends only on intrinsic properties and not on the particle's speed or radius of the circle it moves in.
In complex motions such as helical paths, cyclotron frequency helps predict the rotating aspect of the motion. As seen with harmonic solutions of the motion's differential equations, \( x(t) = R \cos(\omega t + \phi_x)\) and \( y(t) = R \sin(\omega t + \phi_y)\), illustrate the periodic aspect at a natural rate dictated by \(\omega\), a hallmark of cyclotron movement.