Equation (31.9) says that ${\mathit{v}}_{\mathbf{a}\mathbf{b}}\mathbf{=}\mathit{L}\mathit{d}\mathit{i}\mathbf{>}\mathit{d}\mathit{t}$(see Fig. 31.8a). Using Faraday’s law, explain why point a is at higher potential than point b when i is in the direction shown in Fig. 31.8a and is increasing in magnitude. When i is counterclockwise and decreasing in magnitude, is${\mathit{v}}_{\mathbf{a}\mathbf{b}}\mathbf{=}\mathit{L}\mathit{d}\mathit{i}\mathbf{>}\mathit{d}\mathit{t}$ still correct, or should it be ${\mathit{v}}_{\mathbf{a}\mathbf{b}}\mathbf{=}\mathbf{-}\mathit{L}\mathit{d}\mathit{i}\mathbf{>}\mathit{d}\mathit{t}$? Is${\mathit{v}}_{\mathbf{a}\mathbf{b}}\mathbf{=}\mathit{L}\mathit{d}\mathit{i}\mathbf{>}\mathit{d}\mathit{t}$ still correct when i is clockwise and increasing or decreasing in magnitude? Explain.

Whenever a conductor is placed in a varying magnetic field, an electromotive force is induced. The induced emf in a coil is equal to the rate of change of flux linkage.

$\epsilon =-L\frac{di}{dt}$where $\epsilon$is induced emf, Lis inductance.

When variable current (i) passes through inductor, an emf is induces according to faraday’s law

$\epsilon =-L\frac{di}{dt}$.

The point a is at higher potential than b because as the current increases$\frac{di}{dt}$ is positive and induced emf is opposite to flow of current making a at higher potential.

$v_{ab}=L\frac{di}{dt}$.

$v_{ab}=L\frac{di}{dt}$this expression is also correct when I is flowing counterclockwise and decreasing in magnitude because$\frac{di}{dt}$ will be negative but since$v_{ab}$ will be in same direction as in current to support its decrement (as per faraday’s law).

By the same reason,$v_{ab}=L\frac{di}{dt}$ is valid when current is either increasing or decreasing in clockwise direction.

Thus, point a is at higher potential than point b because current increases making the$\frac{di}{dt}$ positive and induced emf directed opposite to current. Also,$v_{ab}=L\frac{di}{dt}$ is valid when current is flowing counterclockwise or clockwise in either increasing or decreasing manner.