Purchase Solution

# Particle in a magnetic field

Not what you're looking for?

The motion of a charged particle in an electromagnetic field can be obtained from the Lorentz equation for the force on a particle in such a field.
If the electric field vector is E and the magnetic field vector is B the force on a particle of mass m that carries charge q and has velocity v is given by:

F = qE +qv X B

1.

If there is no electric field and the particle enters the magnetic field in a direction perpendicular to the lines of teh magnetic flux, show that the trajectory is a circle with radius

r = mv/qB = v/w

where w=gB/m is the cyclotron frequency

2.
Choose the z-axis to lie in direction of B and let the plane containing E and B be the yz-plane.
Thus:
B = Bk E = E_y j + E_z k

Show that the z-component of the motion is given by:

z(t) = z(0) + (z_dot(0))*t+(qE_z / 2m) * t^2

3.
Continue the calculation and obtain expressions for dx/dt and dy/dt Show that the time average of these velocity components are:

<dx/dt> = E_y/B <dy/dt> = 0

4.
Integrate the velocity equation from (3) and show (with appropriate initial conditions - see attached file) that:

x(t) = -A/w * cos(wt) + E_y/B *t
y(y) = A/w *sin (wt)

Plot the trajectory for the following cases:

A > E_y/B
A = E_y/B
A < E_y/B

##### Solution Summary

The solution is 14 pages long including full explanations and derivation of all equations.

##### The Moon

Test your knowledge of moon phases and movement.

##### Classical Mechanics

This quiz is designed to test and improve your knowledge on Classical Mechanics.

##### Basic Physics

This quiz will test your knowledge about basic Physics.

##### Intro to the Physics Waves

Some short-answer questions involving the basic vocabulary of string, sound, and water waves.

##### Variables in Science Experiments

How well do you understand variables? Test your knowledge of independent (manipulated), dependent (responding), and controlled variables with this 10 question quiz.