CBSE 2025, Set 1

Question

(a) (i) Define self-inductance of a coil. Derive the expression for the energy required to build up a current $I$ in a coil of self-inductance $L$.

(ii) 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.

Edvaya · Accepted Answer

The correct answer is Subjective. **(i) Self-inductance — definition and energy derivation:** **Definition:** The self-inductance (L) of a coil is defined as the magnetic flux linked with the coil when a unit current flows through it. Alternatively, it is the magnitude of the emf induced in the coil when the rate of change of current through it is unity. $$L = \frac{N\phi}{I} \quad 	ext{or} \quad L = \frac{|e|}{dI/dt}$$ **Energy derivation:** When current $I$ builds up in a coil of self-inductance $L$, the back-emf induced opposes the increase. To maintain growth of current, the source must supply power: $$P = |e| \cdot I = L\frac{dI}{dt} \cdot I$$ The work done in building current from 0 to $I$ is: $$dW = P\,dt = LI\,dI$$ $$W = \int_0^I LI\,dI = L\left[\frac{I^2}{2}ight]_0^I = \frac{1}{2}LI^2$$ $$\boxed{U = \frac{1}{2}LI^2}$$ This energy is stored in the magnetic field of the inductor. --- **(ii) Graphs for inductors with same $dI/dt$:** **Graph (I): |emf| vs rate of change of current ($dI/dt$)** $$|e| = L\frac{dI}{dt}$$ Both inductors show linear graphs through origin, but with different slopes: - Inductor 1 (L₁ = 10 mH): slope = 10 mH - Inductor 2 (L₂ = 20 mH): slope = 20 mH ![Graphs of emf vs dI/dt for two inductors](https://www.edvaya.com/images/cbse/physics/2025/set1/q32a-graph-emf-vs-di-dt.webp) **Graph (II): Energy stored vs current** $$U = \frac{1}{2}LI^2$$ Both are parabolic curves through origin, with different values: - Inductor 1: $U = \frac{1}{2}(10 	imes 10^{-3})I^2$ — lower parabola - Inductor 2: $U = \frac{1}{2}(20 	imes 10^{-3})I^2$ — higher parabola (steeper) ![Graphs of energy stored vs current for two inductors](https://www.edvaya.com/images/cbse/physics/2025/set1/q32a-graph-energy-vs-current.webp) > **Key:** For graph (I), the emf-vs-dI/dt plot is linear (inductor 2 has twice the slope of inductor 1). For graph (II), the energy-vs-I plot is parabolic (inductor 2 stores twice the energy at the same current).

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Physics > Magnetism Induction And Alternating Current > Electromagnetic Induction

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