Question

The currents passing through two inductors of self-inductances 10 mH and 20 mH increase with time at the same rate. Draw graphs showing the variation of:
(I) The magnitude of emf induced with the rate of change of current in each inductor:
(II) The energy stored in each inductor with the current flowing through it:

Answer 1

(I) Magnitude of Induced EMF Based on Rate of Current Change:

The induced electromotive force \( E \) in an inductor is determined by the rate at which the current changes, \( \frac{dI}{dt} \), according to the equation:

\[ E = L \frac{dI}{dt} \]

Here, \( L \) represents the inductor's inductance.

Given that the rate of current change \( \frac{dI}{dt} \) is identical for both inductors, the induced EMF is directly proportional to the inductance value.

Consequently, an inductor with greater inductance will exhibit a higher induced EMF. For instance, if one inductor has a self-inductance of 20 mH and another has 10 mH, the EMF in the 20 mH inductor will be double that of the 10 mH inductor.

This proportional difference will be evident in a graph plotting EMF against time.

(II) Energy Stored in Inductors with Current:

The energy \( W \) stored within an inductor is calculated using the formula:

\[ W = \frac{1}{2} L I^2 \]

The variables in this formula are:

• \( W \): the stored energy
• \( L \): the inductance
• \( I \): the current flowing through the inductor

When the current is the same, the energy stored is directly proportional to the inductance. Therefore, the 20 mH inductor stores twice the energy of the 10 mH inductor.

A graph of energy versus current will display a quadratic relationship. For any given current, the curve representing the larger inductance will be consistently higher by a factor of two.