Force on a Moving Charge in Uniform Magnetic and Electric Fields

The force applied to any charged particle due to magnetic and electric fields is known as Lorentz force. Both the fields apply different forces, and the value of the force varies to a large extent. In the case of a magnetic field, the charged particle will feel a force that moves in the direction of the velocity component. It won’t experience any force if it moves parallel due to the present magnetic field. These fields and forces are often studied separately, but the sum of force on a moving charge in a uniform magnetic and electric field is called Lorentz force. 

Body 

•     Force due to magnetic field 

The motion of the charges is always responsible for causing a magnetic field or magnetic force on that charge. It is often stated that the two charges that have the same amount of charge and are moving in similar directions develop an attractive magnetic force between them. At the same time, the two charges that move in opposite directions develop repulsive magnetic force between them. Let’s explain the concept with a derivation. 

Explanation: If we consider two charged and moving objects, they will have some amount of magnetic force developed between them. However, the direction of the force will always depend on the charge that each of the objects possesses. One of the easy ways to find the magnetic force that develops between the two charges is by assuming that a constant amount of charge, q, is moving with some constant velocity, v. The magnetic force, in this case, is assumed to be B. The relation between velocity and magnetic fields is that they always work perpendicularly. Here is how we write the formula of the magnetic force due to a magnetic field imposed on a charged particle. 

Fm = qv× B

Where q is the charge, B is the magnetic field, v is the velocity and θ is angle between magnetic field and velocity. Remember, they always form a cross product and act perpendicular to each other. Here, the velocity and magnetic field form a cross product that can be represented by:

Fm = qvBsinθ

To find the direction of the magnetic field, we always use Fleming’s right-hand thumb rule. As per the rule, the fingers of the right hand are stretched in such a form that the thumb, center finger, and forefinger are in perpendicular directions to each other. In such stretching, the direction of the forefinger points to the direction of the magnetic field if the thumb points in the direction of the conductor’s motion and the center finger points in the direction of the induced current. 

•     Force due To electric field 

When an electric charge is imposed on the moving charges, the objects present experience the force due to the electric field. The force on electric charges is always in a parallel or antiparallel direction to the electric field. Unlike magnetic force, electric force is capable of doing work and can impart kinetic energy to the system. 

 Fe = qE

Where Fe is the force due to the electric field, q is the charge present, and E is the electric field. 

•     What is the Lorentz force? 

In simple terms, the force that is executed by both electric and magnetic fields on the moving charges is known as Lorentz force. We may also say that the force on a moving charge in uniform magnetic and electric fields is known as Lorentz force. Here is how we represent the whole force: 

F = Fm + Fe

F = qE + q(v × B)

In the above equation, the first term, qE, is contributed by the electric field, and the second term is due to the presence of magnetic fields. 

F = q (E + v × B) 

 where F is the total force the system experiences, q is the charged particle, v is the velocity with which the charge moves, E is the electric field, and B is the magnetic field. In case of any charged moving particles, the magnetic force depends on the magnitude of the charge and the cross product of v and B. 

Here are the two main properties of the Lorentz force. 

Case 1 – Suppose a case where electric and magnetic fields are uniform, and the direction of the electric field, magnetic field, and velocity of the charged particle is parallel to one another. In such a case, the magnetic force due to the magnetic field will be zero. 

Fm = qv × B = qvB sin 0 = 0

Hence, the charge, in this case, will possess a rectilinear motion, and the field present will be only an electric field.

Case 2 – When both the forces on a moving charge in uniform magnetic and electric fields are parallel to each other but perpendicular to the velocity of the particle, the force due to the magnetic field won’t be zero, meaning, Fm ≠ 0.

Hence, in this case, the moving charged particle will have a circular motion. 

Conclusion  

To explain the major effects of forces that act on a particular charged particle, we use the concept of Lorentz force. The right-hand thumb rule gives the direction of the magnetic force. One thing to remember in the force on a moving charge in a uniform magnetic and electric field is that the magnetic force has to make the charged particle move in a circular direction while the electric field is used for the acceleration of the charged particle as well as to impart the needed kinetic energy to the charged particle.