Question

A 20 Henry inductor coil is connected to a 10 ohm resistance in series as shown in figure. The time at which rate of dissipation of energy (joule's heat) across resistance is equal to the rate at which magnetic energy is stored in the inductor is

A) $\eqalign{
  & \dfrac{2}{{\ln 2}} \cr
  & \cr} $
B) $ln2$
C) $2ln2$
D) $\dfrac{1}{2}\ln 2$

Answer 1

Hint: In this L-R circuit, equate both rate of magnetic energy stored in inductor and rate of dissipation of energy across resistance. Simplifies both the equation so that we take out the time. Substitute L and R values and calculate it until we get a final answer.

Complete step by step solution:
For L-R circuit, we take
$i = \dfrac{E}{R}\left( {I - {e^{\dfrac{{ - t}}{t}}}} \right)$
Where $L = \dfrac{L}{R}$
Rate of magnetic energy stored in inductor = $i \times L \times \dfrac{{di}}{{dt}}$ -----(1)
Rate of dissipation of energy across resistance =${i^2}R$
We both equate the equation.
$\eqalign{
  & {i^2}R = iL\dfrac{{di}}{{dt}} \cr
  & R \times \dfrac{E}{R}\left( {I - {e^{\dfrac{{ - Rt}}{L}}}} \right) = \left( {L \times \dfrac{E}{R} \times \dfrac{R}{L}{e^{\dfrac{{ - Rt}}{L}}}} \right) \cr} $
$\eqalign{
  & \Rightarrow 2{e^{\dfrac{{ - Rt}}{L}}} = 1 \cr
  & \Rightarrow \dfrac{R}{L}t = \ln 2 \cr
  & \Rightarrow t = \dfrac{L}{R}\ln 2 \cr
  & \Rightarrow t = \dfrac{{20}}{{10}}\ln 2 \cr
  & \therefore t = 2\ln 2 \cr} $

Hence, the given correct answer C.

Additional information:
Resistance is defined as a measure of the opposition to current flow in an electrical circuit, resistance is measured in ohms.
Current is nothing but the rate at which electrons flow past a point in a complete electrical circuit.
Potential difference is nothing but is the difference in the amount of energy that charge carriers have between two points in a circuit, it is measured volt.
Magnetic field means it contains energy, also called magnetic energy. Because a magnetic field is generated by electric currents the magnetic energy is an energy form of moving charge carriers.
Dissipation of energy means dissipation is a term that is often used to describe ways in which energy is wasted. And any energy that is not transferred to useful energy stores is said to be wasted because it is lost to the surroundings for example we see electrical cables warming up.

Note:
Here, we use the rate of magnetic energy stored in the inductor. It means, the energy stored in a magnetic field is equal to the work needed to produce a current through the inductor. Rate of dissipation of energy across resistance means as a charge q moves through a resistor, it loses a potential energy qV where V is the potential drop across the resistor.