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

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

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

5. In this piece of conductor there are

6. In this piece of conductor there are NAL free electrons

7. 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

8. 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

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

10. 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

11. 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

12. 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

13. Substituting for I gives

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

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

16. but L/t is

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

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

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

20. 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

21. If the particle moves at angle q to the field

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

23. 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

24. 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 =

25. 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