7.5.3

# Moving Charges in a Magnetic Field

Test yourself

## Force on a charged particle

A magnetic field will create a force on a charged particle. This is called the Lorentz force.

### Lorentz force

• The Lorentz force always acts perpendicular to the velocity of a particle.
• This means that the force acts as a centripetal force.
• Particles in magnetic fields undergo circular motion.

### Calculation

• We know that the centripetal force is given by
• ${F_c}=mv^2/r$
• If the particle is travelling in a magnetic field perpendicular to the field lines it must be acted upon by the Lorentz force
• $F=qvB$

### Calculation 2

• If we equate the two forces we get
• $qvB=mv^2/r$
• We can then rearrange to get the radius of the circle
• $r=mv/qB$

## Particle Accelerators

A cyclotron is a method of accelerating particles.

### Cyclotrons

• A cyclotron is made up of two semi-circular plates separated by a gap with a potential difference across it.
• A magnetic field is applied so that it is perpendicular to the semi-circles.
• Charged particles follow a circular path parallel to the plates due to the magnetic field.

### Accelerating

• A positive particle reaches the gap every half rotation.
• The voltage drop accelerates the particle towards the negative terminal across the gap.
• The voltage across the gap is varied on every half rotation so that the particle speed is always increased.

• The radius of the circular motion is given by
• $r=mv/qB$
• As the velocity of the particle increases, the radius of its circular motion increases.
• The particle spirals out of the machine at a very high speed.