A closed loop of wire is given the shape of a circle with a...

A closed loop of wire is given the shape of a circle with a radius of $0.500 \mathrm{~m} .$ It lies in a plane perpendicular to a uniform magnetic field of magnitude $0.400 \mathrm{~T}$. If in $0.100 \mathrm{~s}$ the wire loop is reshaped into a square but remains in the same plane, what is the magnitude of the average induced emf in the wire during this time?

Key Concepts

Faraday's Law of Electromagnetic Induction

This law states that a changing magnetic flux through a closed conducting loop induces an electromotive force (emf) in the loop. The induced emf is proportional to the rate of change of the flux, and its direction is given by Lenz's law, which ensures that the induced current opposes the change in flux.

Magnetic Flux

Magnetic flux is defined as the product of the magnetic field strength, the area through which the field lines pass, and the cosine of the angle between the field and the perpendicular to the area. It quantifies the total number of magnetic field lines passing through a given surface and is a key factor in determining the induced emf.

Conservation of Perimeter in Reshaping

When a closed loop of wire is reshaped, its total length remains constant. This conservation of the perimeter means that even if the shape changes, the length of the wire (or the boundary of the loop) is fixed. This constraint is important when determining the new area of the loop after reshaping from one geometric configuration to another.

Geometric Properties of Closed Shapes

The calculation of areas for different geometric shapes, such as circles and squares, is essential when the loop changes shape. Even with a constant perimeter, the area enclosed by different shapes is generally not the same. Understanding the relationship between perimeter and area for various shapes allows one to determine how the magnetic flux changes as a result of the reshaping.

Transcript

00:01 Hi in the given problem first of all there is a circle made up of a wire whose radius is given as 0 .500 meter so its area area of this circle initial area will be given by pi r square now circumference of this circle or we can say length of the wire used to make the circle is 2 pi into r meter is this is 2 pi into 0 .50 meter which will come out to be pi meter so then it is re -shaped into a square loop now the perimeter of square which is 4 times the side length suppose the side length is a small so that will also be equal to length of this wire.

01:12 Hence the side of the square will be l by 4.

01:17 Means this is pi 5 4 meter.

01:21 Hence area of the square loop now will be a 2 is equal to square of the side.

01:28 Means this is 5 by 4 to the whole square meter square.

01:36 And here it was 5 to 0 .4 to the square which will come out to be 0 .25 which may also be written as pi by 4 meter square so that was the initial area and this is the final area we can write it like pi square by 16 meter square also the magnitude of magnetic field linked through this loop perpendicularly that is 0 .4000 s law and this change in the shape of the loop from circle to square takes place in a time 0 .1 0 .0 second so using paradise laws of electromagnetic induction emi the emf induced the average emf induced will be given by by negative of time rate of change of magnetic flux which is linked through this loop.

02:47 It can also be written as final magnetic flux minus initial magnetic flux divided by time.

02:54 So here this is for 5 -2, this is b into a2 into cos 0 degree as loop is perpendicular to the magnetic field, means magnetic field vector is parallel to the area vector...

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