Magnetic Force

Magnets are magnificent objects that create an invisible area of magnetic force around them; this area is known as the magnetic field. 

When an object is brought near to this field, it gets attracted to the magnet without any physical contact. So, magnetism is the property of an object of getting attracted to the magnet.

On bringing the N-pole of two freely suspended magnets together, the two magnets repel each other; however, on reversing the direction of one magnet, both magnets attract each other.

Materials like nickel, iron show magnetism, while materials like wool, cotton don't.

What is Magnetic Force?

We know that the current in the conductor is due to the motion of free electrons in a definite direction. When such a conductor/wire is placed in the magnetic field, each electron moving in this field experiences a force. Thus the current-carrying wire/conductor experiences a force in the magnetic field.

Now, let’s say a charged particle with velocity ‘v’ is moving in a magnetic field ‘B’. Because of the interaction between the magnetic field produced by moving charge and the magnetic field applied, the charge ‘q’ experiences a force called the magnetic force.

Magnetic Force

Now, let’s understand how to find magnetic force.

Let’s suppose that a positive charge ‘q’ moving in a uniform magnetic field Bwith a velocity \[\overrightarrow{v}\]. The angle between \[\overrightarrow{B}\] and \[\overrightarrow{v}\] is θ The force experienced by this charged particle depends on the following factors:

• The magnitude of the force experienced by the charge is directly proportional to its magnitude, i.e., F ∝ q….(1)

• The magnitude of the force \[\overrightarrow{F}\] is directly proportional to the velocity component \[\overrightarrow{v}\] acts in a perpendicular direction to the magnetic field, i.e.,

                       F  ∝ v Sin θ…..(2)

• The magnitude of force \[\overrightarrow{F}\] is directly proportional to the magnitude of the magnetic force applied. It is given by:

                      F ∝ B….(3)

Combining equations eq (1), (2), and (3), we get:

                                 F ∝ q B Sin θ

Now, removing the sign of proportionality constant, we get:

                        F = k q v B Sin θ

Here, k is a constant whose value is 1, so the equation becomes:

                  F =  q v B Sin θ

                \[\overrightarrow{F}\]       = q ( \[\overrightarrow{v}\] x \[\overrightarrow{B}\] )

Definition of \[\overrightarrow{B}\]

If v = 1, q = 1, and Sin θ = 1,then:

          F = B

It means that the magnetic field induction at a point on the magnetic field is equal to the force experienced by the unit charge moving with a unit velocity in a direction perpendicular to that of the magnetic field at that point.

Now, we will look at the special cases on determining the direction of magnetic force:

Direction of Magnetic Force

We can see that the direction of the magnetic force is the cross-product of velocity and magnetic field, which is perpendicular to the plane containing \[\overrightarrow{v}\] and \[\overrightarrow{B}\]. 

Let’s consider a piece of paper where \[\overrightarrow{v}\] and \[\overrightarrow{B}\]are in the plane of the paper, then according to the Right-handed screw rule, the direction of \[\overrightarrow{F}\] on the positively charged particle will be perpendicular to the plane of this paper upwards. However, on the negatively charged particle, the direction changes to downward. 

We can see the visual representation of the right-hand screw or right-hand rule for both positive and negatively charged particle below:  

(Image to be added soon)

Special Cases on Magnetic Force

We know that the equation for the magnetic force is given by:

             F = qvB Sin

Now, we will consider the following cases by taking ‘' as the variable here:

Case 1:

If θ = 0° or 180°, then Sin θ= 0

From the above equation, we get F = 0. It means a charged particle is moving in a direction parallel to that of the magnetic field. At this moment, a charge experiences no force.

Case 2:

If θ = 90°, then Sin θ = 1

From the above equation, F = 1

At this moment, a charged particle is moving in along a line perpendicular to the direction of the magnetic field. At this moment, a charge experiences a maximum force.

At times, when a charge experiences a maximum force, the direction of the force can be determined by Fleming’s left-hand rule. Now, let’s understand what this rule says:

Fleming’s Left-Hand Rule

Fleming’s left-hand rule is applicable for a special case when a charge experiences a maximum force. 

If we stretch our first two fingers and the thumb, where the first finger is an index finger while the second one is the middle finger. 

The index finger, central finger, and the thumb lie perpendicular to each other, and each of these represent a direction of a few things; let’s see what are those:

• Index/First finger - Represents - the direction of the magnetic field

• Central finger - Represents - the direction of motion of a moving charge (electric charge)

• Thumb - Represents - force experienced by the charged particle

If the velocity \[\overrightarrow{v}\] is along the X-axis and \[\overrightarrow{B}\] is along the Y-axis, then \[\overrightarrow{F}\] is along the Z-axis. Below is the representation of this statement:

(Image to be added soon)