Question

In the circuit shown in the figure. The coil has inductance $L$ and resistance $R$. When point $X$ is joined to $Y$, the time constant is $\tau$ during the growth of current. When the steady state is reached, heat is produced in the coil at a rate of $P$. Now, point $X$ is joined to $Z$, then, the total heat produced when current in the circuit reduced to zero is

• A. $P\tau$
• B. $\frac{1}{2}P\tau$
• C. $2P\tau$
• D. $\frac{3}{2}P\tau$

Answer 1

The correct option is B $\frac{1}{2}P\tau$


Steady-state current in the circuit when $Y$ remains connected to $X$,

$i_0=E/R$

So, heat is produced at the rate,

$P=i_0^{2}R=\frac{E^2}{R}$

After point $X$ is connected to $Z$ and the current reduces from its maximum value to zero, all the stored energy in the inductor will be dissipated by its resistive property.

Energy stored in the coil when $Y$ remains connected to $X$,

$=\frac{1}{2}L{i}_0^2=\frac{1}{2}L{\left(\frac{E}{R}\right)}^2$

$=\frac{1}{2}\left(\frac{L}{R}\right)\left(\frac{E^2}{R}\right)$

$=\frac{1}{2}\tau P$

Hence, option $\left(B\right)$ is the correct answer.