GATE 2018 EC Q.36: solution and answer

GATE 2018 · ECQ.36 · 2 marks · NAT

For the circuit given in the figure, the magnitude of the loop current (in amperes, correct to three decimal places) 0.5 second after closing the switch is ________.

GATE 2018 EC question 36: A single series loop drawn as a rectangle. — Figure from the GATE 2018 EC paper. It is part of the question.

Answer and solution

ANSWER

The loop current 0.5 s after the switch closes is $i (0.5) = 0.5 (1 - e^{- 1}) \approx 0.316 A$ .

Official answer: 0.284 to 0.348Source: Official answer key, GATE 2018 EC (IIT Guwahati)

Take $t = 0$ as the instant the switch closes. Before that the loop is open, so no current flows and $i_{L} (0^{-}) = 0$ . An inductor's current cannot change instantly, so $i (0^{+}) = i_{L} (0^{-}) = 0$ .
Once the switch closes, the circuit is a single series RL loop. Total resistance $R = 1 + 1 = 2 Ω$ , source $V = 1 V$ , $L = 1 H$ .
Final value: in DC steady state the inductor behaves as a short circuit, so $i (\infty) = \frac{V}{R} = \frac{1}{2} = 0.5 A$ .
Time constant: $τ = \frac{L}{R} = \frac{1}{2} = 0.5 s$ .
First-order response: $i (t) = i (\infty) + [i (0^{+}) - i (\infty)] e^{- t / τ}$ , which gives $i (t) = 0.5 (1 - e^{- 2 t}) A$ for $t \geq 0$ .
At $t = 0.5 s$ , exactly one time constant: $i (0.5) = 0.5 (1 - e^{- 1}) = 0.5 \times 0.63212 = 0.31606 A$
Correct to three decimal places, the loop current is $0.316 A$ .

KEY IDEA

A switched series RL loop with a DC source is a first-order circuit. The current starts at zero, because an inductor's current cannot jump. It rises towards $V / R_{total}$ with time constant $τ = L / R_{total}$ , and both resistors count because the whole loop is in series. Here 0.5 s is exactly one time constant, so the current has reached $1 - e^{- 1} \approx 63.2%$ of its final value.

WATCH OUT FOR

Using only one of the 1 Ω resistors. Because the second resistor sits on the far side of the loop, it is easy to miss that it is in series too. With $R = 1 Ω$ you get $τ = 1 s$ and $i (0.5) = 1 - e^{- 0.5} \approx 0.393 A$ , which is outside the key. A second slip is to use $τ = R L$ or $τ = R / L$ instead of $L / R$ , or to report the final value $0.5 A$ as if the transient had already died away.

From the GATE 2018 EC paper. Every question of this paper · Download the full paper · All 20 Transients questions, with notes

GATE 2018 EC Q.36: solution and answer

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