Question

A closed circuit consists of a source of constant emf $xi$ and a choke coil of inductance $L$ connected in series. The active resistance of the whole circuit is equal to $R$. It is in steady state. At the moment $t=0$ the choke coil inductance was decreased abruptly $4$ times its initial value. The current in the circuit as a function of time $t$ is $\xi /R[1+x{e}^{-4tR/L}]$. Find out value of $x$_______

Answer 1

Let us say, inductance reduces by $\eta$ times the initial value and $\xi$ be the emf applied

We write the equation of the circuit as, $Ri+\frac{L}{\eta }\frac{di}{dt}=\xi$

for $t\ge 0$.

The current at $t=0$ just after inductance is changed, is $i=\eta \frac{\xi }{R}$, so that the flux through the inductance is unchanged.

We look for a solution of the above equation in the form
$i=A+B{e}^{-t/C}$

substituting $C=\frac{L}{\eta R},B=\eta -1,A=\frac{\xi }{R}$ gives,

$i=\frac{\xi }{R}(1+(\eta -1\left)e^{-\eta Rt/L}\right)$.

By putting $x=3$, the above equation becomes equation given in the question.

Accepted answers : $3,3.0,3.00$