Question

a)

Find ε(R,t), the electromotive force (EMF) around a loop that is at distance R from the z axis, where R is restricted to the region outside the iron rod as shown. Take the direction shown in the figure as positive.

Express ε(R,t) in terms of A, B₀, B₁, R, and any needed constants such as ,π, and μ₀

b)

Due to the cylindrical symmetry of this problem, the induced electric field can depend only on the distance R from thez axis, where R is restricted to the region outside the iron rod. Find this field.

Express in terms of quantities given in the introduction(and constants), using the unit vectors in the cylindricalcoordinate system,,, and.

Answer 1

Concepts and reason

The concept used to solve this problem is magnetic flux and electromotive force.

Initially, the electromotive force or induced emf around the loop can be calculated by determining the rate of change of magnetic flux. Later the induced electric field can be calculated by using the Faraday’s law in one dimension.

Fundamentals

The expression for the electromotive force as a function of time is,

Here, is the induced emf as a function of time, is the change of magnetic flux and is the change in time.

The expression for the magnetic flux is,

Here, B is the magnetic field and A is the area of cross section of the loop of coil.

According to the faraday’s law the expression for the electric field is,

Here, is the induced electric field and is the elementary length.

(a)

The variation of uniform magnetic field inside the rod can be written as,

The expression for the electromotive force is,

Substitute for to find

Substitute for .

(b)

The expression for the induced electric field is,

Substitute for and for .

Substitute for .

Rearrange the above expression for .

Ans: Part a

The electromotive force around the loop is .

Part b

The induced electric field is..