Question

A magnetic field directed into the page changes with time according to

B = 0.069 0t² + 1.40, where B is in teslas and t is in seconds. The field has a circular cross section of radius

R = 2.50 cm (see figure below).

When t = 3.20 s  and r₂ = 0.020 0 m, what is the magnitude of the electric field at point P₂?

Answer 1

Concepts and reason

The concepts required to solve this problem are Faraday’s law, and magnetic flux.

Initially, find the relation between the electric field and magnetic field by using the concept of faraday’s law and magnetic flux.

Then, substitute the values in the expression of obtained electric field to find the value of the electric field at a radius .

Fundamentals

The faraday’s law for a non-conservative electric field can be written as follows:

Here, is the non-conservative electric field, ds is the line integral element, is the rate of change of magnetic flux.

The magnetic flux can be written as follows:

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

The magnetic field region is circular such that the induced electric field is symmetric, which means that the electric field is equal in magnitude at all points on the circle of radius , containing point .

The faraday’s law for a non-conservative electric field can be written as follows:

Since, the electric field is symmetric, it can be treated as a constant and can be taken out of the integral. The line integral of the circle integrates to form the circumference of the circle.

The circumference of the circle can be written as follows:

Here, is the radius of the circle.

Solve the integral as follows:

Substitute for in the above expression.

Now, the area vector of the circle is out of the page and the magnetic field is directed into the page. This means that the angle between the magnetic field and the area vector is 180 degrees.

The magnetic flux can be written as follows:

Substitute 180 degrees for in the above expression.

Substitute -BA for in the expression .

The area of the circle of radius can be written as follows:

Substitute for A in the expression .

Substitute for B in the above expression and solve for

Substitute 0.02 m for and 3.2 s for in the above expression.

Ans:

The electric field is .