5.0 Induced Electric Field

Consider a conducting loop is placed in rest in a constant magnetic field. At $t = 0$ magnetic field start to change with time. Hence, magnetic flux is changing. We know that when magnetic flux is changing through a closed conducting loop current is induced in loop which opposes to change in magnetic field. At $t = 0$ there was no current in loop we can say that free electrons were at rest (consider average random motion is zero). After $t = 0$ current start to flow because of changing magnetic field. We know that magnetic field cannot exert force on electrons at rest. Thus magnetic force cannot start the induced current. Electrons can be force to move only and only by electric field. So this electrical field is induced by changing magnetic field not by charged particles according to Coulomb’s law or Gauss’s law. This electric field is neoconservative or nonelectrostatic in nature. We cannot define electrical potential corresponding to this field. We called it induced electric field.

The work done per unit charge to move it $d\vec l$ is $\vec E.d\vec l$. Induced emf in conducting loop is

$$\varepsilon = \oint {\vec E.d\vec l} $$

Using Faraday’s law of induction,

$$\varepsilon = - \frac{{d\Phi }}{{dt}}$$

or, $$\oint {\vec E.d\vec l} = - \frac{{d\Phi }}{{dt}}$$

The line of this induced electric field is curved. There is no starting or ending point of induced electrical field.

Example: The magnetic field $B$ is directed into the plane of paper. $PQRP$ is a semicircular conducting loop of radius $r$ with the centre at $O$. The loop is made to rotate clockwise with a constant angular velocity $\omega $ about an axis passing through $O$ and perpendicular to the plane of the paper. The resistance of the loop is $R$. Obtain an expression for the magnitude of the induced current in the loop. Plot a graph between the induced current $i$ and $\omega t$, for two periods of rotation.

When the loop rotates through an angle THETA, which is less than PAI, the area inside the field is

$$A = \frac{\theta }{\pi }\frac{{\pi {r^2}}}{2} = \frac{{\theta {r^2}}}{2} = \frac{{\omega t{r^2}}}{2}$$

The flux of the magnitude field at time $t$ is

$$ = BA = \frac{{\omega t{r^2}}}{2}$$

The induced emf = $$ - \frac{{d\Phi }}{{dt}} = - \frac{{B\omega {r^2}}}{2}$$

The magnitude of the induced current will be

$$i = \frac{{B\omega {r^2}}}{{2R}}$$

As the flux is increasing, the direction of the induced current will be anticlockwise so that the field due to the induced current is opposite to the original field. After half rotation, the area in the field region will be given by

$$A(t) = \frac{{\pi {r^2}}}{2} - \frac{{\omega t{r^2}}}{2}$$

Hence, the induced current will have the same magnitude but opposite sense. The plot for two time periods is shown in figure.