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# Lorentz Force

## Lorentz Force Meaning

Last updated date: 22nd Mar 2023
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Since the beginning of human evolution, we knew about electric fields and magnetic fields and had an assumption that both magnetic and electric fields are independent of each other. As the advancements took place through experiments it was more evident that both magnetic and electric fields are interrelated with each other. Hendrik Antoon Lorentz, a dutch physicist, was the first to bring up the theory in order to explain the relationship between electric and magnetic fields. The Lorentz force explains the relation between magnetic and electric fields.

Since the beginning of human evolution, we knew about electric fields and magnetic fields and had an assumption that both magnetic and electric fields are independent of each other. As the advancements took place through experiments it was more evident that both magnetic and electric fields are interrelated with each other. Hendrik Antoon Lorentz, a dutch physicist, was the first to bring up the theory in order to explain the relationship between electric and magnetic fields. The Lorentz force explains the relation between magnetic and electric fields.

## Lorentz Force Formula

The study of the magnetic fields is done by comparing the effects of electric fields with the effect of magnetic fields. Whenever we study the magnetic field we should keep in mind that the magnetic field is associated with moving charges, which means all the fields, forces that we derived for a point charge in a static condition will not be in good agreement with the charge considered in a magnetic field.

Charge under motion will result in current, then in order to derive the force acting on the moving charge, we will analyze the magnetic effect on electric current and hence derive the Lorentz force Formula.

### Lorentz Law

We know that every charge experiences force when it is under the effect of either electric field or magnetic field. Dutch physicist Hendrik Antoon Lorentz, in the year 1895, formulated the formula for force giving rise to both electric and magnetic field effects.

### What is Lorentz Force Law? Define Lorentz Force:

Lorentz Force law is defined as the combined force experienced by a point charge due to both electric and magnetic fields.

According to the Lorentz force definition, the Lorentz forces are the forces on moving charges due to the electromagnetic fields. The Lorentz force equation is given by small derivation.

### Explain Lorentz Force

Consider a charge q moving with velocity v and it is moving in the existence of both electric and magnetic fields. Then we write:

The force due to the electric field is given by = F$_{E}$ = qE

The force due to the magnetic field is given by = F$_{B}$ = q(v х B)

Where,

q - Charge on particle under observation

E - Electric field due to point charge

v - Velocity of moving charges

B - The magnetic field due to moving charges

Now, Formula of Lorentz force is given by,

⇒F$_{L}$ = F$_{E}$ + F$_{B}$

⇒F$_{L}$ = qE + q(v х B)

⇒F$_{L}$ = q{E + (v х B)} ………..(1)

Equation (1) is known as the Lorentz force equation. The direction of Lorentz force is perpendicular to the direction of the moving charge and the magnetic field. The Lorentz force direction is well explained by using the Right-hand rule (Lorentz Force right-hand rule).

### Properties of Lorentz Force:

Case 1:

If the Electric field, magnetic field, and the direction of the velocity of the particle are parallel to each other and E and B are uniform,

then, F$_{B}$ = qv sin 0 = 0

Therefore, the charge will perform the rectilinear motion, because the charge will be accelerated due to the electric field.

Case 2:

If the Electric field and magnetic field are parallel to each other, and the direction of the velocity of the particle is perpendicular to  E and B,

then, F$_{B}$ ≠ 0

Therefore, the charge will perform the circular motion, because the charge will be accelerated due to the electric field.

### Example:

1: What Should be the Velocity of a Charged Particle so that it will Not Experience Any Force or it will Not be Accelerated?

Ans:

For a charged particle to remain unaccelerated, it must satisfy the condition that electrostatic force and magnetic force are equal.

⇒ for a = 0, Then, F$_{E}$ = F$_{B}$

Then,

⇒ F$_{E}$ = F$_{B}$

⇒ qE = q(v х B)

⇒ E = vB sinθ

The angle between the magnetic field and the velocity of the charged particle is 90⁰.

Then,

⇒ E = vB

⇒ v = $\frac{E}{B}$

Therefore, for the charged particle to remain accelerated the velocity of the charge must be equal to the ratio of the magnitude of the Electric and magnetic field.

### Did You Know?

Lorentz force explains the importance of the effects of force acting on a charged particle. The right-hand rule is easy to calculate the magnetic force as the direction of the force can be visualized and demonstrated given by Lorentz force law.