The Magnetic Field

The force on a charge q moving with a velocity The magnitude of the force

left-hand rule right-hand rule

Motion in magnetic field 1) Uniform, 2) Uniform, 3) Nonuniform

The angular velocity

The angular velocity (cyclotron frequency ) does not depend on velocity! The force is always perpendicular to velocity, so it cannot change the magnitude of the velocity, only its direction. The work done by the magnetic force is zero! Motion of a charged particle under the action of a magnetic field alone is always motion with constant speed.

Exercise 1 An electron, q= C moves with velocity It enters a magnetic field with What is the force on the electron?

Using Crossed and Fields Velocity selector independent of the mass of the particle!

Mass spectrometer

Thomson’s e/m experiment 1897: Cavendish Laboratory in Cambridge, England

Electron motion in a microwave oven A magnetron in a microwave oven emits electromagnetic waves with frequency f=2450 MHz. What magnetic field strength is required for electrons to move in circular paths with this frequency?

Problem 5 Hall effect: The magnetic force on the charge carries in a wire can be used to determine their sign. Show that there will be an electric field, set up inside a wire in a magnetic field, that is perpendicular to the direction of the current. You should be able to show that the sign of the field depends on whether the mobile charges are positive or negative.

You place a slab of copper, 2.0 mm thick and 1.5 cm wide, in a uniform magnetic field with magnetic field with magnitude 0.40 T. When you run a 75-A current in the +x direction, you find by careful measurement that the potential at the left side of the slab is 0.81  V higher than at the right side of the slab. From this measurement, determine the concentration of mobile electrons in copper.

Exercise 3 A wire of length l and mass m is suspended as shown. A uniform magnetic field of magnitude B points into the page. What magnitude and direction would a current, passing through a wire, have to have so that the magnetic and gravitational forces would cancel?

Problem 4 A metal wire of mass m can slide without friction on two parallel, horizontal, conducting rails. The rails are connected by a generator which delivers a constant current i to the circuit. There is a constant, vertical magnetic field, perpendicular to the plane of the rails. If the wire is initially at rest, find its velocity as a function of time. i B l generator

The angular velocity

Uniform magnetic field,

Uniform, When a charged particle has velocity components both perpendicular and parallel to a uniform magnetic field, the particle moves in a helical path. The magnetic field does no work on the particle, so its speed and kinetic energy remain constant.

Example: A proton ( ) is placed in the uniform magnetic field directed along the x-axis with magnitude T. Only the magnetic force acts on the proton. At t=0 the proton has velocity components Find the radius of the helical path, the angular speed of the proton, and the pitch of the helix (the distance traveled along the helix axis per revolution).