*Magnetic Force on moving charge in magnetic field is possible only due to the presence of electric field created by the charges moving from positive end to the negative end.*Now let’s see how a magnetic fields are created.

**When there is current flowing due to the motion of electric charges produce magnetic fields. When the nucleus of atom orbits continuously then the magnetic force on moving charges in magnetic field is determined.**The force on a negative charge is in **exactly the opposite direction** to that on a positive moving charge.

Fundamentally, when a current is passed over an element electric field is witnessed Here we consider the element to be a solenoid which in turn create __magnetic field__s in and around the region. This magnetic force exerted on the charge particle will affect any particle entering the field.

It is a known __fact that an electric field__ is produced by static charges and when another charged particle is brought closer it is either attracted or repelled. So in this way, the electric field has a force that will act on the charges present in the electric field.

Similarly, there exists a magnetic force on a moving charge in magnetic fields. Here we will deal with inductors to show how the force on a moving charge in magnetic field is possible. In an electric field, it is capacitors that will be the reason for force on a moving charge.

**How charged particle moves in magnetic field**

We consider a current filament where electric current flows in a certain direction of the magnetic field is produced in circular form. This filament can also be a solenoid.

At this moment a charge enters the magnetic field region with a certain velocity. Since the magnetic field lines are not similar to the __electric field line__, they will form a circular path. The charge entering the magnetic field will travel in the circular path as well.

Force on a moving charge on the magnetic field is the process happening basically when a charge passes through the magnetic flux lines. The magnetic fields exert forces by the magnetic flux line on a charge moving within becomes zero if it is parallel to magnetic field lines.

**Force on moving charge formula**

We are well aware of how the magnetic fields exert forces on a moving charge that comes inside the flux lines. The magnetic field is in right angles to the charge that is undergoing motion.

**The magnetic force on moving particle in the magnetic fields is denoted by : F = q V B sineθ.**

Lets try to understand the derivation and implementation :

Because the charge does not experience any change in its kinetic energy, since the charged particle moves in a circular motion. So when anything that experiences a circular motion will have zero displacements and the kinetic energy will remain the same.

Considering this we shall determine the formula for magnetic force on a moving charge in magnetic field.

The __magnetic flux__ line or the magnetic fields are denoted by the letter B, the charge that enters and moves inside the magnetic field is denoted by the letter q. The velocity with which the charge moves inside the magnetic field is denoted by the letter V.

When the charge moves inside the magnetic field, the field exert forces on the charge. This magnetic force is related to certain parameters. The parameter for the force magnitude is as explained; it is proportional to the magnitude of the charge, the magnitude of the velocity of the charge under motion, and the magnetic field.

The exert forces by magnetic field proportional to sine θ. Meaning, θ is the angle made by the velocity of the charge that moves with the magnetic flux lines.

**Force on moving charge formula (Explanations):**

Now the formula for magnetic force on moving charge is **F = q V B sineθ.**

As in the case of force it is basically a vector quantity having magnitude and direction. The formula mentioned previously is used to calculate magnitude of the force. The direction of the magnetic force is the direction of the charge moving in the magnetic field.

The direction of the magnetic charge travelling inside the magnetic field is in right angles to both the velocity and the magnetic field. The formula for this condition is **F = q V B sine θ an**.

Therefore when the motion of the charge is right angles to the __velocity and the magnetic field__ the formula is revised and given as **F = q (V X B)**. Because θ becomes 90⁰ and sine 90⁰ is equal to one.

Hence the formula for the magnetic force on moving charge in the magnetic field is given by three different conditions and can be used according to the problems provided.

**Moving charge in a uniform magnetic field derivation**

A Uniform magnetic field is produced when a current-carrying solenoid is passed with an electric current. This is easily explained using Right Hand Thumb Rule or also called as Lorentz Force.

The above mentioned formula is used to calculate the magnetic force employed on the charge moving inside the magnetic field.

The right-hand thumb rule is defined as; the thumb indicating the direction of velocity, the index finger indicating the direction of the magnetic field (B), and the middle finger indicating the direction of the resultant force.

The right-hand thumb rule is also known as Lorentz Force. The formula of Lorentz Force is **F = q V B sine θ**. Here q is the charge in the magnetic field, V is the velocity, B is the magnetic field and θ is the angle made between velocity and magnetic field.

**Zero force on moving charge:**

It is now a known fact that the charge moving inside the magnetic field will undergo a circular motion. The force acting on this will have a different result compared to the conventional one.

**When electric current is present in a solenoid, eventually a magnetic is created. The flux lines are in a circular motion that is they are produced around the solenoid.**

Hence when a charge moves inside the region of the magnetic field they follow the direction of the magnetic flux lines. A circular motion is eventually created inside. The charge will move in the same direction and then have no change in the kinetic energy.

Since there is no displacement in the whole system the force is said to be zero. The reason is that the charge will go on and on moving in circles in the direction of the magnetic flux lines.

The velocity with which the charge moves inside the magnetic field is parallel to the magnetic field. So the magnetic force on moving charge will be eventually zero.

A charge moving equally parallel in the same direction of the magnetic field, then magnetic force acting that particular magnetic field is zero.

In essence of the work done on the charge in a magnetic field is zero or minimum. In a magnetic field if the kinetic energy of the charge is said be zero the the system obeys work-energy theorem.

Here in this case when the magnetic force becomes perpendicular to the velocity the direction might not change but the magnitude will change. So the word done on the charge will be zero, making the force acting on the charge also zero.

## Direction of moving charge in magnetic field:

The magnetic field direction created by a moving charge is perpendicular to the direction of motion of the charged particle. Hence the magnetic force generated due to a magnetic field is perpendicular to the direction of motion of the movement and speed of the charged particle.

**Problems and solutions:**

**Problem 1: Consider a charge to move in the north direction with a speed of 3 x 106 m/s. a magnitude of 4.0T will act in the west direction. Now calculate the magnitude of the force on moving charge in the magnetic field? [The charge moving inside the magnetic field is the proton].**

Solution:

Let us consider the right-hand thumb rule. The force coming out of the hand is the magnetic force on moving charge.

__Magnitude of the force__ is **F = q V B sineθ**

F = (1.6 x 10-19C x 3 x 106 x 4 T x sine 90⁰)

**F = 1.92 x 10-12 N**

**Problem 2: Calculates the earth’s magnetic field when the positive moving charge in the system has a velocity 2 x 105m/s moving in the north direction and the magnitude of the force acting on it is 1.2 x 10-13N in the west direction.**

Solution:

Formula is **F = q V B sine θ**

B = F / (q x V x sine θ)

B = (1.2 x 10-13) / (2 x 105 x 1.6 x 10-19 x sine 90⁰)

**B = 3.75 T**

Therefore now it is clear that the magnetic force on moving charge has different conditions from the explanation and the formulae.

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Hi…I am Keerthana Srikumar, currently pursuing Ph.D. in Physics and my area of specialization is nano-science. I completed my Bachelor’s and Master’s from Stella Maris College and Loyola College respectively. I have a keen interest in exploring my research skills and also have the ability to explain Physics topics in a simpler manner. Apart from academics I love to spend my time in music and reading books.

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