CBSE 2025 · Region 5 · Set 1 · Q32 · 5 marks
(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 $\displaystyle 10$ mH and $\displaystyle 20$ 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.(i)Define the term mutual inductance. Deduce the expression for the mutual inductance of two long coaxial solenoids of the same length having different radii and different number of turns.(ii)The current through an inductor is uniformly increased from zero to $\displaystyle 2$ A in $\displaystyle 40$ s . An emf of $\displaystyle 5$ mV is induced during this period. Find the flux linked with the inductor at $\displaystyle \mathrm{t}=10 \mathrm{~s}$.
(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 $\displaystyle 10$ mH and $\displaystyle 20$ 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.
(i)
Define the term mutual inductance. Deduce the expression for the mutual inductance of two long coaxial solenoids of the same length having different radii and different number of turns.
(ii)
The current through an inductor is uniformly increased from zero to $\displaystyle 2$ A in $\displaystyle 40$ s . An emf of $\displaystyle 5$ mV is induced during this period. Find the flux linked with the inductor at $\displaystyle \mathrm{t}=10 \mathrm{~s}$.
Marking-scheme solution
(a)
Self-inductance is the magnetic flux linked with a coil when the current through the coil is unity.
Self-inductance is the emf induced in the coil when the rate of change of current through the coil is unity.
To maintain growth of current, power has to be supplied from an external source:
$\displaystyle P=e I$
$\displaystyle \frac{d W}{d t}=L I \frac{d I}{d t}$
$\displaystyle d W=L I d I$
$\displaystyle W=\int L I d I$
$\displaystyle =\frac{1}{2} L I^{2}$
(I)
$\displaystyle E=-L \frac{d I}{d t}$
(II)
$\displaystyle U=\frac{1}{2} L I^{2}$ — parabolic graph obtained.
($\displaystyle 1$ indicates $\displaystyle 10$ mH) & ($\displaystyle 2$ indicates $\displaystyle 20$ mH)
(b)
Mutual inductance is defined as the emf induced in the primary coil when the current in the secondary coil changes at unit rate.
Mutual inductance is defined as the magnetic flux linked with the primary coil when the current in the secondary coil is unity.
Consider two long co-axial solenoids each of length l. Radius of inner solenoid $\displaystyle S_{1}$ is $\displaystyle r_{1}$ and number of turns per unit length is $\displaystyle n_{1}$. The corresponding quantities for outer solenoid $\displaystyle S_{2}$ are $\displaystyle r_{2}$ and $\displaystyle n_{2}$ respectively. Let $\displaystyle N_{1}$ and $\displaystyle N_{2}$ be the total number of turns of coils $\displaystyle S_{1}$ and $\displaystyle S_{2}$ respectively.
When a current $\displaystyle I_{2}$ is set up through $\displaystyle S_{2}$, it sets up magnetic flux through $\displaystyle S_{1}$:
$\displaystyle N_{1} \phi_{1}=M_{12} I_{2}$
$\displaystyle =\left(n_{1} l\right) \times\left(\pi r_{1}^{2}\right) \times\left(\mu_{0} n_{2} I_{2}\right)$
$\displaystyle =\mu_{0} n_{1} n_{2} \pi r_{1}^{2} l I_{2}$
$\displaystyle M_{12}=\mu_{0} n_{1} n_{2} \pi r_{1}^{2} l=M_{21}$
(ii)
$\displaystyle e=L \frac{d I}{d t}$
$\displaystyle L=\frac{e}{\dfrac{d I}{d t}}$
$\displaystyle =\frac{5 \times 10^{-3}}{\dfrac{2}{40}}$
$\displaystyle =0.1 \mathrm{H}$
$\displaystyle \phi=L I$
$\displaystyle =0.1 \times \frac{2}{40} \times 10$
$\displaystyle =0.05 \mathrm{~Wb}$
Electromagnetic InductionInductanceApplylong_answerhard
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CBSE Class 12 Physics past-paper question from the 2025board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.