Force acting on a charged particle moving through a magnetic field © D Hoult 2008

Consider a conductor of length L, having n free electrons per unit volume. A current, I is flowing through it

Consider a conductor of length L, having n free electrons per unit volume. A current, I is flowing through it

In this piece of conductor there are

In this piece of conductor there are NAL free electrons

In this piece of conductor there are NAL free electrons If all these electrons pass through end x in time t then the current, I is given by

In this piece of conductor there are NAL free electrons If all these electrons pass through end x in time t then the current, I is given by n A L e t

If there is a magnetic field of flux density B at 90° to the current, the conductor will experience a force of magnitude

If there is a magnetic field of flux density B at 90° to the current, the conductor will experience a force of magnitude I L B

If there is a magnetic field of flux density B at 90° to the current, the conductor will experience a force of magnitude I L B This is the sum of the forces on all the electrons, so the force F acting on each electron is given by

If there is a magnetic field of flux density B at 90° to the current, the conductor will experience a force of magnitude I L B This is the sum of the forces on all the electrons, so the force F acting on each electron is given by I L B I B F = = n A L n A

Substituting for I gives

Substituting for I gives n A L e B F = = t n A

Substituting for I gives n A L e B L e B F = = t n A t

but L/t is

but L/t is the (drift) velocity of the electrons

but L/t is the (drift) velocity of the electrons therefore

but L/t is the (drift) velocity of the electrons therefore F = e v B

In general the magnitude of the force acting on a charged particle moving with velocity v, at 90° to a magnetic field of flux density B, is given by F = q v B where q is the charge on the particle

If the particle moves at angle q to the field

If the particle moves at angle q to the field the magnitude of the component of its velocity at 90° to the field is

If the particle moves at angle q to the field the magnitude of the component of its velocity at 90° to the field is v cos a

If the particle moves at angle q to the field the magnitude of the component of its velocity at 90° to the field is v cos a = v sin q Therefore, in general F =

If the particle moves at angle q to the field the magnitude of the component of its velocity at 90° to the field is v cos a = v sin q Therefore, in general F = q v B sin q