Motion in a Magnetic Field

Motion of a Charged Particle in Magnetic Field

The interaction of the electric field and magnetic field and the motion of charged particles in the presence of both the electric and magnetic fields and also have derived the relation of the force acting on the charged particle,

A force acting on a particle is said to perform work when there is a component of the force in the direction of motion of the particles

We have a charged particle carrying a charge q moving in a uniform magnetic field of magnitude B, the magnetic force acts perpendicular to the velocity of the particle.

We say that no work is done by the magnetic force on the particle, and hence, no change in the velocity of the particle can be seen. Mathematically, when the velocity of the particle v is perpendicular to the direction of the magnetic field, we can write,

The magnetic force is directed towards the center of circular motion undergone by the object and acts as a centripetal force. 

Thus, if v and B are perpendicular to each other, the particle describes a circle.

 When a component of velocity is present along the direction of the magnetic field B, then its magnitude remains unchanged throughout the motion, as no effect of a magnetic field is felt upon it, the motion due to the perpendicular component of the velocity is circular in nature.

As the radius of the circular path of the particle is r, the centripetal force acting perpendicular to it towards the center can be given as,

F=mv2r

Also, the magnetic force acts perpendicular to both the velocity and the magnetic field and the magnitude can be given as,

F=qvB

Equating the two, we get

F=mv2r =qvB

Or r=mvqB

Here, r gives the radius of the circle described by the particle. Also, if we write the angular frequency of the particle as ω, then we can write,

v=ωr

So, ω=2πv=qBm

Here, v is the frequency of rotation of the particle. The time for one revolution can be given as,

T=2πω=1v

The distance moved by the particle along the direction of the magnetic field in one rotation is given by its pitch. p=vpT=2πmvpqB

Where vp is the velocity parallel to the magnetic field.

Motion in Combined Electric and Magnetic Fields:

An electric field is created by the electrical charge particle and is perpendicular to the magnetic field.

 The unit of the electric field is Volt/meter and is a vector quantity.

The electric field is conservative in nature.

The magnetic field is the space around the magnet, where the magnetic force is observed. This field is created by the movement of the electric charges.

Magnetic lines represent the direction of the magnetic field. The electric fields are generated around the particles, which are characterised by electric charges.

 The force created by the electric field is much stronger than the force created by the magnetic field. The orbiting motion of charges in a magnetic field is the basis for measuring the mass of an atom. A closed loop is formed by the magnetic field lines, and the electric field lines do not form a loop.

Lorentz Force

When a moving charge comes under the influence of magnetic and electric fields, it experiences Lorentz force. Lorentz force is a kind of force that can be calculated as the vector sum of forces created by magnetic and electric fields.

Fnet = FE + FB

Fnet = q(E + v x B)

The below figure shows the representation of the electric field and the magnetic field along with the motion of charge when they are perpendicular to each other.

When the values of E and B are adjusted such that the magnitude of the two forces are equal, the total force acting on the charge is zero, and the charges will move in the field undeflected.

When the strength of electric and magnetic fields are varied to get the forces due to electric and magnetic fields to be equal (FE = FB), then the charge can move in the field without any deflection.

Fnet = 0

FE = FB

qE=Bqv

E=Bv

v=E/B

This case is used when charged particles of a certain velocity (E/B) are used to pass through the crossed fields undeflected. This phenomenon is called a velocity selector. It was applied by J. Thomson to evaluate the charge to the mass ratio in 1897.

Velocity selector:

Velocity selector is also known as Wien filter is a device with a perpendicular arrangement of electric and magnetic fields which is used as a velocity filter.

 

Velocity selector exploits the principle of motion of a charge in a uniform magnetic field. According to this principle, the force experienced by a moving charge with speed v in a uniform magnetic field is given as:

F = Bqv

Where,

F: force experienced by the charge

B: magnetic field

q: moving charge

v: speed of the charge

      The speed of charged particles in velocity selector is

v=EB

Fields of the velocity selector

Uniform electric field: This field is generated by the top plate with negative charges and the bottom plate with the positive charges. These charges result in the formation of the field which is pointing in the upward direction in the figure.

Uniform magnetic field: This field is present uniformly between the two charged plates such that it can be directed either inwards or outwards.

Limitations of the velocity selector

Neither the mass nor the charge of the particles is considered before passing through the filter.

All the uncharged particles pass through the filter.

It is used in mass spectrometers, where charged objects are distinguished as per their charge to mass ratio.

Cyclotron

A cyclotron is a machine used to accelerate charged particles or ions to high energies.

To enhance the energies of charged particles, cyclotron uses magnetic as well as electric fields. It is called crossed fields since the magnetic and electric fields are perpendicular to each other.

 A cyclotron consists of two flat and hollow semicircular metal disc-like containers represented as D1 and D2. These containers are separated by a narrow gap. Disc-like containers connected to a high-frequency oscillator have the capacity to generate an alternating voltage.

Alternating electric field is created between D1 and D2 as well from D2 to D1. The charged particle in the containers is accelerated when it passes through the gap between the metal containers.

When a charged particle moves into a hollow semicircular metal disc-like container, it moves in a circular path and with a constant speed. Due to the shielding effect, the electric field is zero inside the container.

 The frequency of the cyclotron is independent of the speed of the charged particle and the radius of the circular path. These machines are used to bombard nuclei and are used to study nuclear reactions.