Question

The magnitude of the induced electric field in the orbit at any instant of time during the time interval of the magnetic field change is
(A) $\dfrac{BR}{4}$
(B) $\dfrac{BR}{2}$
(C) BR
(D) 2BR

Answer 1

Hint: We need to assume that the radius of the orbit remains constant and then try to solve the problem. The application of magnetic field induces an EMF in the orbit. The induced emf is defined as the work done by an induced electric field in moving a unit positive charge around a closed loop. So, by making use of the following concepts we will try to move ahead with the problem.

Formula Used:
For an induced electric field, we have,
$\int{E.dr=A\dfrac{dB}{dt}}$

Complete answer:
Let the electric field be E and the magnetic field be given by B.
Let us consider a charge q moving in a circular orbit of radius R with an angular velocity. A uniform magnetic field switched on, which increases from 0 to B in one second.
We have the change in magnetic field at a given time interval is given as $\dfrac{dB}{dt}$=B
The magnitude of the induced electric field at an instant of time during the time interval of the magnetic field change is given by
$\int{E.dr=A\dfrac{dB}{dt}}$
$\Rightarrow E\int{dr=A.B}$
$\Rightarrow $ E (2$\pi $ R) = $\pi {{R}^{2}}B$, where the area A is given by $\pi {{R}^{2}}$
and R is the constant radius of the orbit.
$\Rightarrow E=\dfrac{BR}{2}$
Thus the correct option for the induced electric field in the orbit at any instant of time during the time interval of the magnetic field change is given as $\dfrac{BR}{2}$.

Hence the correct option is (B) $\dfrac{BR}{2}$.

Additional Information:
Electromagnetic induction is the production of an electromotive emf in a conductor as a result of a changing magnetic field about the conductor.

Note:
The parameters that are necessarily very important in electromagnetic induction are magnetic field, inductor or coil and the third is the movement of the coil with respect to the field. The key consideration in this problem is the changing magnetic field with time.