Question

A rectangular coil with resistance R has N turns, each of length and width w as shown in Figure P31.29. The coil moves into a uniform magnetic field with constant velocity . What are the magnitude and direction of the total magnetic force on the coil for the following situations? (Use the following as necessary: N,B, w, v, and R.)

Figure P31.29

(a) The coil enters the magnetic field.
F =   [Direction?]
(b) The coil moves within the field.
F =   [Direction?]
(c) The coil leaves the field.
F = [Direction?]

Answer 1

Concepts and reason

The concepts required to solve the given question are magnetic force, Ohm’s law and the relation between the magnetic flux and the induced emf.

First, find an expression for the magnetic force due to a current carrying wire when the coil enters the magnetic field. Then, use Ohm’s law and the Faraday’s law and express the force in terms of the given quantities. In the next step, find the direction of the magnetic force using the right hand rule.

Next, find an expression for the magnetic force when the coil moves within the field from the relation between the magnetic force, current, and the magnetic flux.

After that, find the direction of the magnetic force when the coil is completely in the magnetic field.

In the next part, find the expression for the magnetic force when the coil leaves the field. Finally, find the direction of the magnetic force by applying the right hand rule.

Fundamentals

The force that arises due to the motion of charged particle within the magnetic field is known as the magnetic force.

The magnetic force due to a current carrying wire is given by,

Here, is the magnetic force, is the magnetic field, is the current, is the length of the wire and is the angle between the velocity vector and the magnetic field vector.

In this case, the velocity vector and the magnetic field vector are perpendicular to each other. Hence the angle is.

Thus the magnetic force expression becomes,

For number of turns, the magnetic force is,

According to Faraday’s law of electromagnetic induction is “the rate of change of flux linkage is equal to the induced emf”.

Here, is the magnitude of the induced emf, is the number of turns, and is the rate of change of magnetic flux.

Ohm’s law gives the relation between current, voltage and resistance. The law states that the “electric current is directly proportional to the voltage and inversely proportional to the resistance of the circuit”.

The expression for Ohm’s law is given by,

Here, V is the voltage, I is the current and R is the resistance of the circuit.

(a.1)

Consider the expression for the magnetic force due to a current carrying coil,

…… (1)

According to the Faraday’s law, the induced emf in the circuit is,

…… (2)

The magnetic flux induced in the coil is,

Here, is the magnetic flux, is the number of turns, is the area of the coil, is the angle between the magnetic field and the area vector and is the magnetic field.

In this case, the magnetic field and the area vector are parallel to each other. Hence the angle becomes.

Thus the magnetic flux is,

The area of the rectangular loop is,

Here, is the length and is the width of the loop.

Use for in terms of .

…… (3)

Substitute the equation (3) in equation (2).

Using the relation for the velocity of the coil moves in a magnetic field,

…… (4)

Here, v is the velocity of the coil, is the change in length, and is the time taken.

Substitute equation (4) in the above equation,

…… (5)

The expression for the current flows in the circuit is given by,

Substitute equation (5) in the expression for the current in the circuit.

Substitute the expression for the current in equation (1).

(a.2)

Right hand rule states that, place the right hand thumb in the direction of the magnetic field and the curled fingers gives the direction of current.

In this case, there is an induced emf due to the change in magnetic flux. Thus a magnetic field is generated in the opposite direction of the applied field. It will be pointing out of the page. By applying the right hand rule, the direction of current is counterclockwise.

Now curl the fingers in a direction rotating the current in to the magnetic field. Then thumb gives the direction of the magnetic force.

In this case, the magnetic force is acting towards left.

(b.1)

When the coil moves within the field, the flux linked is given by,

Here, the flux is constant.

Hence there is no induced emf in the coil, so that .

The current in the coil is,

Thus the magnetic force when the coil moves within the field is,

(b.2)

Magnetic flux linked with the coil when it moves within the magnetic field is constant. Hence the change in magnetic flux is zero.

Then, there is no induced emf in the coil also the current in the coil is zero. Thus, the magnetic force is zero. Hence there is no direction for the magnetic force.

(c.1)

As the coil leaves the magnetic field, the rate of change of magnetic flux decreased.

The decreasing rate of change of magnetic flux is,

The induced emf is given by,

Substitute for .

The current flowing through the loop is,

Substitute for .

The magnetic force acting on the coil is,

Substitute for and for .

(c.2)

Here, the rate of change of magnetic flux is decreased. Hence the direction of the induced emf is clockwise. But the current through the portion of the coil in the magnetic field is upwards. Then the magnetic force is acting towards the left.

Ans: Part a.1

The expression for the magnetic force acting on the coil when it enters to the magnetic field is .

Part a.2

The direction of magnetic force when the coil enters to the magnetic field is towards left.

Part b.1

The total magnetic force when the coil moves within the magnetic field is zero.

Part b.2

There is no direction for the magnetic force acting on the coil when it moves within the magnetic field.

Part c.1

The magnetic force acting on the coil when it leaves the magnetic field is .

Part c.2

The direction of magnetic force on the coil when it leaves the magnetic field is towards left.