Magnetic Fields

Chapter 26

26.2 The force exerted by a magnetic field

Definition of B

26.3 Motion of a charged particle in a magnetic field

Applications

A circulating charged particle

Crossed fields: discovery of the electron

The cyclotron and mass spectrometer

26.4 Magnetic force on a current-carrying wire

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This lecture

The definition of B

BvqF

The sign of q matters!

Magnetic force and field

Charged particle moving in a plane perpendicular to a uniform magnetic

field (into page).

Find expression for radius, r

CHECKPOINT: Here are three situations in which a charged particle with velocity v travels through a uniform magnetic field B.

In each situation, what is the direction of the magnetic force FB on the particle? A. LeftB. UpC. Into pageD. RightE. DownF. Out of page

Answers: (a) +z (out)

(b) –x (left, negative particle)

(c) 0

CHECKPOINT: The figure shows the circular paths of two particles that travel at the same speed in a uniform B, here directed into the page. One particle is a proton; the other is an electron.

(a) Which particle follows the smaller circleA. pB. e

(b) Does that particle travel A. clockwise or B. anticlockwise?

Answers: (a) electron (smaller mass)

(b) clockwise

pe

Crossed magnetic and electric fieldsNet force:

The forces balance if the speed of the

particle is related to the field strengths by

qvB = qE

BvqEqF

v = E/B (velocity selector)

Measurement of q/m for electronJ J Thomson 1897

EXERCISE: Find an expression for q/m

Sun-to-aurora TV analogy

9

A small part of the sky overhead

CHECKPOINT: the figure shows four directions for the velocity vector v of a positively charged particle moving through a uniform E (out of page) and uniform B.

(a) Rank directions A(1), B(2) and C(3) according to the magnitude of the net force on the particle, greatest first.

(b) Of all four directions, which might result in a net force of zero:A(1), B(2), C(3) or D(4)?

Answers:

(a) 2 is largest, then 1 and 3 equal (v x B = 0)

(b) 4 could be zero as FE and FB oppose

EXAMPLE: The magnetic field of the earth has magnitude 0.6 x 10-4 T and is directed downward and northward, making an angle of 70° with the horizontal. A proton is moving horizontally in the northward direction with speed v = 107 m/s.

Calculate the magnetic force on the proton by expressing v and B in terms of components and unit vectors, with x-direction East, y-direction North and z-direction upwards).

Picture the problem:

Velocity vector is in the y-direction.

B is in the yz plane

Force on proton must be towards West, ie in negative x-direction

Circular motion of a charged particle in a magnetic field

It was invented in 1934 to accelerate particles, such as protons and deuterons, to high kinetic energies.

S is source of charged particles at centre

Potential difference across the gap between the Dees alternates with the cyclotron frequency of the particle, which is independent of the radius of the circle

The Cyclotron

Schematic drawing of a cyclotron in cross section. Dees are housed in a vacuum chamber (important so there is no scattering from collisions with air molecules to lose energy).

Dees are in uniform magnetic field provided by electromagnet.

Potential difference V maintained in the gap between the dees, alternating in time with period T, the cyclotron period of the particle.

V creates electric field in the gap, but no electric field within the dees, because the metal dees act as shields.

Particle gains kinetic energy q V across gap each time it crosses

Key point: fosc= f = qB/2m is independent of

radius and velocity of particle

The Cyclotron

EXAMPLE: A cyclotron for accelerating protons has a magnetic field of 1.5 T and a maximum radius of 0.5 m.

(a) What is the cyclotron freqency? (b) What is the kinetic energy of the

protons when they emerge?

26.4 Magnetic force on a current-carrying wire

Wire segment of length L carrying current I. If the wire is in a magnetic field, there will be a force on each charge carrier resulting in a force on the wire.

Flexible wire passing between pole faces of a magnet.

(a) no current in wire

(b) upward current

(c) downward current

26.4 Magnetic force on a current-carrying wire

EXERCISE: A wire segment 3 mm long carries a current of 3 A in the +x direction. It lies in a magnetic field of magnitude 0.02 T that is in the xy plane and makes an angle of 30° with the +x direction, as shown. What is the magnetic force exerted on the wire segment?