Question

Att=0, a battery is connected to a series arrangement of a resistor and an inductor. At what multiple of the inductive time constant will the energy stored in the inductor’s magnetic field be 0.500its steady-state value?

Short answer

$t=1.23\tau$

Step-by-step solution

Given

Energy stored in the inductor’s magnetic field is 0.500 of its steady state value.

Understanding the concept

Here we have to use the formula for inductor’s magnetic field and current through RL circuit to find the ratio of time and inductive time constant.

Formula:

$$\begin{gathered} i=i_0\left(1-e^{-\left(\frac{t}{\tau }\right)}\right) \\ {U}_B=0.5L{i}^{-2} \end{gathered}$$

Calculate At what multiple of the inductive time constant will the energy stored in the inductor’s magnetic field be 0.500 its steady-state value 

The current through the RL circuit is asfollows:

$i=i_0\left(1-e^{-\left(\frac{t}{\tau }\right)}\right)$

The energy due to inductance magnetic field is

$U_B=0.5L{i}^2$

For steady state,

$U_b^{\text{'}}=0.5L{i}_0^2$

Now we are given that

$$\begin{gathered} U_b=0.5U_b^{\text{'}} \\ {\left(1-e^{-\left(\frac{t}{\tau }\right)}\right)}^2=0.5 \\ 1-e^{-\left(\frac{t}{\tau }\right)}=\sqrt{0.5} \\ {e}^{-\left(\frac{t}{\tau }\right)}=1-\sqrt{0.5} \\ \frac{t}{\tau }=-In\left(1-\sqrt{0.5}\right) \\ \frac{t}{\tau }=1.23 \\ t=1.23\tau \end{gathered}$$