Chapter 27: Charged Particles

The direction of conventional electric current is the direction of flow of positive charge. When the electrons are moving, the conventional current is regarded as flowing in the opposite direction.

Magnetic force on moving charge

The size of force on the moving charge in a uniform magnetic field depends on:

1. The magnetic flux density B
2. The charge Q on the particle
3. The speed v of particle

If the motion is at right angles to the magnetic field:

F = BQv

If the motion is at an angle θ to magnetic field;

F = BQv sinθ

If the equations F = BIL and F = BQv are consistent with one another;

Since current I is rate of flow of charge;

I = Q/t

Substituting,

F = BQL/t

Now L/t is the speed so,

F = BQv

For an electron, with a charge –e the magnitude of force is;

F = Bev

The force on moving charge is sometimes called Bev force.

Orbiting Charge

When a charged particle moves at right angles to a uniform magnetic field, magnetic force F is always perpendicular to its velocity, hence F acts as a centripetal force (force directed towards the center of the circle);

 Centripetal force = mv²/r

The centripetal force is provided by the magnetic force Bev. Therefore

Bev = mv²/r

Cancelling;

R = mv/Be

The equation when rewritten in terms of momentum p of the particle is;

P = Ber

 The charge to mass ratio of an electron

 It involves finding the charge to mass ratio e/m known as specific charge on the electron.

 e/m_e = v/Br

B and r are measurable but v is not;

eV_ca = ½ m_ev²

Where m_e is the electron mass and v is speed

e/m_e = 2V_ca/r²B²

Hall Effect

The hall-effect is another mechanism in which the mechanism in which the electric and magnetic forces in the moving charged particle is balanced.

The production of voltage across a conductor when a current flows through the conductor at right angle to magnetic field. The charge detected by a small voltage across the probe is hall voltage. The greater the magnetic flus density, the greater the hall voltage.

The electric field strength E is related to the Hall voltage V_H:

E = V_H/d

Equating electric and magnetic forces

eE = Bev

Substituting,

eV_H/d = Bev

By substituting v from I = nAve where A is cross sectional area of conductor and n is number density of conducting particles;

eV_H/d = BeI/nAe

V_H = Bid/nAe

V_{H =} BI/nte

V_H = BI/ntq

Positive charged particles will be deflected in the opposite direction to negative charge and so we determine whether the charge particle are negative or positive by using hall voltage.