Figure 30-25 shows a circular region in which a decreasing uniform magnetic field is directed out of the page, as well as four concentric circular paths. Rank the paths according to the magnitude of $\mathbf{\oint }\overrightarrow{\mathbf{E}}\mathbf{.}\overrightarrow{\mathbf{dS}}$evaluated along them, greatest first.

The circular region has decreasing uniform magnetic field directed out of the page.

The magnitude of the induced emf depends upon the magnitude of the changing magnetic flux through the area of the loop. The emf is related to the induced electric field in each part of the loop.

Formulae are as follows:

$\epsilon =\oint \overrightarrow{E}.\overrightarrow{dS}$,

Where,

$\epsilon$= electric flux,

E = electric field,

dS = differential surface.

According to Faraday’s law, changing magnetic flux induces current and emf in the loop. The emf is related to the electric field produced in each part of the loop as,

$\epsilon =\oint \overrightarrow{E}.\overrightarrow{dS}$.

The loops c and d are outside the region having a magnetic field. So, even if the area enclosed is different, the net magnetic flux through both paths is the same. Hence, the induced emf will be the same for both paths.

Path b is having a larger radius than that of path a. Thus, the magnetic flux through the area of b s is more than that through the area of a. Hence, the emf induced for path b is more than a.

In other words, path a will be ranked last as the magnitude of the emf induced in the path is the least.

Hence,the rank of the paths according to the magnitude of$\epsilon =\oint \overrightarrow{E}.\overrightarrow{dS}$ is: d and c tie, b and then, a.

The magnitude of the induced emf is decided by the magnitude of the changing magnetic flux through the area of the loop. Thus, the area of the loop decides the emf induced, and hence, the ranking of the path.