Chapter 2: Problem 53
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
Step by step solution
Write Down the Force Equation
Express the Cross Product
Breakdown into Component Equations
Convert to Equations of Motion
Solve the Equations of Motion
Describe the Motion
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Charged Particle Motion
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
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
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: \(m \frac{d^2 z}{dt^2} = q E\)
Cyclotron Frequency
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.