Question

A circular loop of flexible iron wire has an initial circumference of 164cm , but its circumference is decreasing at a constant rate of 11.0cm/s due to a tangential pull on the wire. The loop is in a constant uniform magnetic field of magnitude 1.00T , which is oriented perpendicular to the plane of the loop. Assume that you are facing the loop and that the magnetic field points into the loop. Find the magnitude of the emf induced in the loop after exactly time 4.00s has passed since the circumference of the loop started to decrease AND find the direction of the induced current in the loop as viewed looking along the direction of the magnetic field. **Please explain each step to help me understand how to solve the problem as well.

Answer 1

Concepts and reason

The concepts used to solve this problem are rate of change of area, Faraday’s law of induction, Lenz’s law, and induced current.

Initially, use the expression for area and circumference of the circle to find the rate of change of area.

Then, use the Faraday’s law to find the magnitude of the induced emf in the loop exactly after time has passed.

Finally, use the Lenz’s law to find the direction of the induced current in the loop as viewed along the direction of the magnetic field.

Fundamentals

The expression for the circumference of the circle is,

Here, the circumference of the circle is and the radius of the circle is .

The expression for the area of the circle is,

Here, is the radius of the circle.

The rate of change of a quantity is defined as the change of the quantity over a time that depends upon the increase or decrease in the value and it can be positive or negative,

“The Faraday’s law of induction states that whenever relative motion exists between a coil and magnetic field, a voltage will induce in the circuit and this voltage is proportional to the rate of change of the flux.”

From the Faraday’s law, the expression for the induced emf is,

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

The expression for the magnetic flux is,

Here, the magnetic flux is , the magnetic field is , the surface area is , and the angle between the normal to the surface and magnetic field is .

The emf in the conducting loop due to the change in the magnetic flux will produce an electric current.

Lenz’s law states that the induced current will be directed against the magnetic field that produces it.

(1)

The expression for the circumference of the circle is,

The expression for the rate of change of circumference is,

Here, the rate of change of circumference is .

Substitute for .

$27=-11.0cm/s $ …… (1)

The expression for the area of the circle is,

The expression for the rate of change of area is,

…… (2)

Here, the rate of change of area is .

Substitute for in the Equation (2).

…… (3)

After , the value of circumference of the loop is decreased by,

Here, the value of circumference of the loop decreased after is .

The expression for circumference of the loop after is,

$C=C-C, $ …… (4)

Here, initial circumference of the loop is .

Substitute for and for .

Substitute for in the above expression.

Rearrange the above expression for and substitute for .

Substitute for in Expression (3).

From the Faraday’s law, the expression for the induced emf is,

…… (5)

The expression for the magnetic flux is,

Substitute for in Expression (5).

Substitute for , for , and for .

Therefore, after the magnitude of the induced emf is .

(2)

The magnetic field is directing into the page.

According to Lenz’s law, the changes produced in the field will be opposed by the induced current.

Hence, it is found that the direction of current will be in clockwise direction.

From the Lenz’s law, it is observed that the current will be in clockwise direction.

Ans: Part 1

After the magnitude of the induced emf is .

Part 2

The induced current will be in clockwise direction.