Question

In the steady state condition, the rate of heat produced in a choke coil is P. The time constant of the choke coil is $\tau $. If now the choke coil is short circuited, then the total heat dissipated in the coil is:
$\begin{align}
  & \text{A}\text{. }\operatorname{P}\tau \\
 & \text{B}\text{. }\dfrac{1}{2}\operatorname{P}\tau \\
 & \text{C}\text{. }\dfrac{\operatorname{P}\tau }{\ln \left( 2 \right)} \\
 & \text{D}\text{. }\operatorname{P}\tau \ln \left( 2 \right) \\
\end{align}$

Answer 1

Hint: A choke coil is an inductance coil of very small resistance which controls the current flow in an ac circuit. Expressions for different physical quantities in a LR circuit can be applied here. Find the expressions for the rate of heat loss and the time constant for the circuit. From these we can find the heat loss.

Complete answer:
Given in the question, that the rate of heat produced in a choke coil is P.
A choke coil produces heat due to resistance it gives to the flow of current. The heat generated by the choke coil can be defined as the loss of energy by the choke coil.
So, we can express the rate of heat loss of the choke coil in terms of the current and resistance as,
$P=i{{R}^{2}}$
Where, i is the current and R is the resistance of the choke coil.
So, we can write,
${{i}^{2}}=\dfrac{P}{R}$
Now, the time constant of a choke coil can be defined as the time required for the current flowing in the circuit to reach its maximum steady state value is equivalent to 5 time constant. It can be mathematically expressed as the ratio of the inductance to the resistance of the circuit.
$\tau =\dfrac{L}{R}$
So, we can write,
$L=\tau R$
The total heat dissipated in the choke coil can be defined as,
$\begin{align}
  & =\dfrac{1}{2}L{{i}^{2}} \\
 & =\dfrac{1}{2}\left( \tau R \right)\left( \dfrac{P}{R} \right) \\
 & =\dfrac{1}{2}P\tau \\
\end{align}$

So, the correct answer is “Option B”.

Note:
A choke coil is used in electrical circuits which allows the DC current to flow through the circuit and blocks the AC part. It can be defined as an inductor which is used to suppress the alternating current without stopping the direct current through it.