# Particles and fields

This page gives a summary of the physics of the motion of charged particles in magnetic and electric fields.

## (a) MAGNETIC FIELDS

When a charged particle of mass m moves in a magnetic field of flux density B at a velocity v there is a force on it. If the field is at right angles to the motion of the particle this force is:

F = Bqv      where q is the charge on the particle

This force is always at right angles to the motion and so the particle is deflected into a circle of radius R.

Therefore:
Magnetic force:      Bqv = mv2/R

Example problem
Example problem A proton (mass 1.66 x 10-27 kg) with a charge to mass ratio (q/m) of 9.6 x 107 Ckg-1 moves in a circle in a magnetic field of flux density 1.2T at 4.5 x 107 ms-1.
Calculate:
(a) the radius of the circle
(b) the kinetic energy of the proton

(a) R = mv2/Bqv = mv/Bq = 4.5 x 107/1.2 x 9.6 x 107 = 0.39 m
(b) kinetic energy = ½ mv2 = 0.5 x 1.66 x 10-27 x (4.5 x 107)2 = 1.68 x 10-12 J = 10.5 MeV

## (b) ELECTRIC FIELDS

A particle of charge +q in an electric field of intensity E experiences a force Eq towards the negative plate.

If the plates are distance d apart and the p.d between them is V volts then E = V/d and the force is given by:

Electrostatic force: Force = Eq = Vq/d

Unlike a magnetic field, in an electric field a charged particle at rest will still feel a force.

Example problem
A proton passes though an accelerating gap in a particle accelerator. If the gap is 0.4 cm wide and the p.d between the two sides is 50 kV calculate:
(a) the force on the proton
(b) the acceleration of the proton

(a) Force = qV/d = 1.6 x 10-19 x 50 x 103/0.4x10-2 = 2 x 10-12 N
(b) Acceleration = F/m = 1.2 x 1015 ms-2

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