2 Motion in a Magnetic Field _ Motion in Combined Electrical and Magnetic Fields

Motion in a Magnetic Field & Motion in Combined Electrical and Magnetic Fields

• When a charged particle having charge q moves inside a magnetic field  with velocity v, it experiences a force

• When is perpendicular to , the force on the charged particle acts as the centripetal force and makes it move along circular path.

• Let m be the mass of charged particle and r be the radius of the circular path.

v and B are at right angles,

• Time period of circular motion of the charged particle is

• Angular frequency,

This is often called cyclotron frequency.

Velocity Selector

• Force in presence of magnetic and electric field,

• Consider that electric and magnetic fields are perpendicular to each other and also perpendicular to the velocity of particle. We have,

• If we adjust the value of and  such that magnitude of the two forces are equal, then total force on the charge is zero and the charge will move in the fields undeflected. This happens when

qE = qvB

• The above condition can be used to select a charged particle of particular velocity from charges moving with different speeds. Therefore, it is called velocity selector.

Cyclotron

• It is used to accelerate charged particles or ions to high energy.
• Electric and magnetic field are used in combination to increase the energy.
• The working of a cyclotron is based on the fact that the frequency of revolution of charged particle is not dependent on the energy.

• Particle moves inside the two semi-circular disc-like metal containers D₁and D₂ called Dees.
• When the particle moves from one Dee to another, it is acted upon by electric field.
• Electric field changes sign alternately. Therefore, the particle is accelerated by the electric field, which increases the energy of the particle.
• The increase in energy increases the radius of the circular path.

Hence, the path is a spiral one.

• Time period of revolution is

This frequency is called the cyclotron frequency.

• When the frequency of applied voltage is equal to cyclotron frequency, it is called resonance condition.
• Since radius of trajectory,

Hence, the kinetic energy of ions is  .