Magnetic and Electric Force on a Point Charge

Introduction

Before we begin the analysis on what is Lorentz force and its applications, we must know about the terms which are familiar to this concept such as magnetism, electricity, velocity and so on. The Lorentz Force was introduced by Hendrik Antoon Lorentz in 1895.

In an electric field a charged particle will always bear a force because of this field. However, charged particles in a magnetic field will only feel a force due to the magnetic field if it is moving relative to this field. These two effects often create a force that we call the Lorentz force.

What is Lorentz Force?

Lorentz force can be defined as the combined effect of magnetic force as well as electric force on a point charge because of the existence of an electromagnetic field. This Force is widely applied in electromagnetism. It can also be called the Electromagnetic force.

What is Lorentz Law?

When a charged particle q travels with a velocity v through a region of space, both fields will exert a force on the particle and the total force on the particle. This is equal to the vector sum of the electric field and magnetic field force.

Lorentz force is explained as per the equation mentioned below,

F = q (E + v.B)

Where,

F = Force appearing on the particle

q = Charge on the particle

E = Electric Field applied on the particle externally

v = Velocity

B= magnetic field available

What is Magnetic Lorentz Force?

The charged particle which travels through an inward magnetic field with certain velocity will carry an electromagnetic force. That force is called the Magnetic Lorentz Force (Fm).

The magnetic force is only applicable when a charged particle is in motion.

So the formula generated for the magnetic field in the scalar form is:

Fm = qv.Bsinθ

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In vector Form,

Fm = Q (vx B)

Magnetic Field Produced by a Moving Charge

Magnetic force is always perpendicular to velocity, so there is no work done on the charged particle. Therefore, the kinetic energy and speed of the particle will remain constant. Here, the direction of motion is affected, but not the speed. This is typically a uniform circular motion. The easiest case appears when a moving charged particle makes an angle of 900 with the magnetic field.

We need to use the right-hand thumb rule to identify the direction of the magnetic field. If a conductor carries current, then the right hand thumb points in the direction of current and curled fingers shows the direction of magnetic field.

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In above fig,

When a charged particle possesses the negative charge which travels across the plane of magnetic field will create a magnetic force that acts perpendicularly to the velocity, and so velocity fluctuates in direction but does not alter magnitude. So the uniform spiral motion comes into action.

As per the Fig given above, the expression can be written as:

B= μ0/4π  (qv . r)/r3

Where,

B = Magnetic field                                               

q = Electric charge

v = Velocity                                               

r = Vector position

μ0 = Magnetic constant

Force due to a Magnetic Field

In the case of magnetic fields, the lines are generated on the North Pole (+) and terminate on the South Pole (-) – as per the below given figure. The curl of a magnetic field generated by a conventional magnet is always positive. The particles which possess the charge will come into view as spiral fields. It will always be perpendicular to the field lines till the particles gain some velocity.

By applying the right hand thumb rule, the force that Lorentz identified in the magnetic field B of a charged particle Q with a velocity v   is acting perpendicularly to the charge as shown in the figure.

The equation for the Force due to magnetic field is

FB = Q (v x B)

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What is the Force Due to Electric Field?

As shown in the Fig. 6, the Force acting on the charge is not dependent upon velocity, but only on electric field. So, force will be applied in the direction of the electric field (Fe).

The field produced by electric current is generated from the positive charge which executes a uniformly situated spiral line that acts outwards. In the case of a negative charge, the direction of the field is reversed. The electric field is directed tangent to the field lines.

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The dimensional factor of electric field is force per unit charge. In the metre - kilogram - second and SI systems, the appropriate units are Newtons per coulomb, and equivalent to volts per metre. The electric field is expressed in units of dynes per electrostatic unit (esu), and is equivalent to stat volts per centimeter.

Fe is the product of charge and electric field.

Mathematically, it can be represented as:

Fe = Q x E

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Electric field effect is based upon the charged particle. If the charge is positive, the electric field will be away from the particle and vice versa.