Question

A coil has Na turns enclosing an area of A. In a physics laboratory experiment, the coil is rotated during the time interval from a position in which the plane of each turn is perpendicular to Earth's magnetic field to one in which the plane of each turn is parallel to the field. The magnitude of Earth's magnetic field at the lab location is B.
a.)What is the total magnitude of the magnetic flux through the coil before it is rotated?
b.)What is the magnitude of the total magnetic flux through the coil after it is rotated?
c.)What is the magnitude of the average emf induced in the coil?

Answer 1

Concepts and reason

The concepts used to solve this problem are magnetic flux and faraday’s law of induction.

Use the relationship between magnetic field, area of the coil, and angle between the magnetic field and the area of the coil to calculate the magnetic flux through the coil before and after it is rotated.

Finally use faraday’s law of induction to calculate the average induced emf in the coil.

Fundamentals

The magnetic flux is a measure of the number of magnetic field lines passing through an area (such as loop of wire).

Expression for the magnetic flux is,

Here, is the magnetic flux, B is the magnetic field, A is the area of the coil, and is the angle between the magnetic field vector and the area vector.

Lenz’s law states that the induced emf of the coil is equal to the negative of the rate of change of magnetic flux times the number of turns in the coil.

Expression for the magnitude ofaverage emf induced in the coil during rotation is,

Here, is the emf of the coil, is the number of loops and is the rate of change of magnetic flux.

(a)

Expression for the initial magnetic flux is,

Here, is the initial magnetic flux before rotation and is the initial angle between the direction of area vector and the magnetic field vector before rotation.

Initially, the plane of the coil is perpendicular to the magnetic field. Since, the direction of area vector is parallel to the magnetic field vector.

Substitute for.

(b)

Expression for the final magnetic flux is,

Here, is the final magnetic flux after rotation and is the final angle between the direction of area vector and the magnetic field vector after rotation.

Finally, the plane of the coil is parallel to the magnetic field. Since, the direction of area vector is perpendicular to the magnetic field vector.

Substitute for.

(c)

Substitute for and for .

The sign is ignored since the magnitude is to be calculated.

Expression for the average induced emf of the coil is,

Substitute BA for in the above expression.

The magnitude gives only the amount of emf induced and not the direction. So, sign is ignored.

Ans: Part a

Total magnitude of the magnetic flux through the coil before it is rotated is

Part b

Total magnitude of the magnetic flux through the coil after it is rotated is

Part c

The magnitude of the average emf induced in the coil is .