An emf of 20 V is applied at time $$t = 0$$ to a circuit containing in series 10 mH inductor and 5 $$\Omega$$ resistor. The ratio of the currents at time $$t = \infty$$ and at $$t = 40$$ s is close to: (Take $$e^2 = 7.389$$)

For a series $$RL$$ circuit the current as a function of time after a steady emf $$E$$ is suddenly applied is obtained from Kirchhoff’s voltage law. Writing the loop equation we get

$$E = iR + L\dfrac{di}{dt}.$$

This is a first-order linear differential equation. Its standard solution is

$$i(t)=\dfrac{E}{R}\Bigl(1-e^{-\dfrac{R}{L}t}\Bigr).$$

First note the two quantities that characterise the circuit:

$$R = 5\;\Omega,\qquad L = 10\;\text{mH}=10\times10^{-3}\;\text{H}=0.01\;\text{H}.$$

Therefore the time-constant $$\tau$$ of the circuit is

$$\tau=\dfrac{L}{R}=\dfrac{0.01}{5}=0.002\;\text{s}.$$ (Only this value will later show why the exponential term is negligibly small.)

The steady-state current, i.e. the current at $$t=\infty,$$ is obtained from the above expression by allowing the exponential term to vanish:

$$i(\infty)=\dfrac{E}{R}\bigl(1-e^{-\infty}\bigr)=\dfrac{E}{R}\times(1-0)=\dfrac{E}{R}.$$

Putting $$E=20\;\text{V}$$ and $$R=5\;\Omega$$ gives

$$i(\infty)=\dfrac{20}{5}=4\;\text{A}.$$

Now we calculate the current at the specified instant $$t = 40\;\text{s}.$$ Using the same general expression,

$$i(40)=\dfrac{E}{R}\Bigl(1-e^{-\dfrac{R}{L}\,40}\Bigr).$$

First evaluate the exponent very carefully:

$$\dfrac{R}{L}\,40=\dfrac{5}{0.01}\times40 =500\times40 =20000.$$

So the exponential term is

$$e^{-20000}.$$

This number is astronomically small. Even a negative exponent of only $$-20$$ already makes $$e^{-20}\approx2.06\times10^{-9};$$ for $$-20000$$ the value is effectively zero in any practical calculation. Hence we write

$$e^{-20000}\approx0,$$

and therefore

$$i(40)=\dfrac{20}{5}\,(1-0)=4\;\text{A}.$$

The ratio asked in the question is

$$\dfrac{i(\infty)}{i(40)} =\dfrac{4\;\text{A}}{4\;\text{A}} =1.$$

Among the numerical alternatives supplied, the value nearest to the exact theoretical ratio $$1$$ is $$1.06.$$ Consequently this option is regarded as the closest correct choice.

Hence, the correct answer is Option A.