The current through two inductors of self-inductance 12 mH and 30 mH is increasing with time at the same rate. Draw graphs showing the variation of the (a) emf induced with the rate of change of current in each inductor (b) energy

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The current through two inductors of self-inductance 12 mH and 30 mH is increasing
with time at the same rate. Draw graphs showing the variation of the
(a) emf induced with the rate of change of current in each inductor
(b) energy stored in each inductor with the current flowing through it.
Compare the energy stored in the coils, if the power dissipated in the coils is the same.

Answers (1)

a) emf induced
$E\, \alpha \, L\frac{di}{dt}$
$E= -L\, \frac{di}{dt}$
$E_{1}= -12\times 10^{-3}\, \frac{di}{dt}$
$E_{2}= -30\times 10^{-3}\, \frac{di}{dt}$

b) Energy stored $V_{0}=\frac{1}{2}\, L\, I^{2}$
$V_{1}=\frac{1}{2}\times 12\times 10^{-3}I^{2}$
$V_{2}=\frac{1}{2}\times 30\times 10^{-3}I^{2}$

$\frac{V_{2}}{V_{1}}= \frac{L_{2}}{L_{1}}\left ( \frac{I_{2}}{I_{1}} \right )^{2}$
Power $p= \frac{dU}{dt}=LI \frac{dI}{dt}$
given $p_{1}= p_{2}$ & $\frac{dI}{dt}$ is constant
$\therefore \frac{I_{1}}{I_{2}}= \frac{L_{2}}{L_{1}}$
$\frac{V_{2}}{V_{1}}= \frac{30}{12}\times \left ( \frac{12}{30} \right )^{2}= \frac{12}{30}= \frac{2}{5}$