Question

What is the voltage across the inductor at $t=0$? (Circuit diagram provided: A 60V voltage source in series with a switch that closes at $t=0$, a 30 ohm resistor, and a 15H inductor.) 

• A. \( 0 \text{ V} \)
• B. \( 20 \text{ V} \)
• C. \( 60 \text{ V} \)
• D. \( 58 \text{ V} \)

Hint:

The key concept here is the behavior of an inductor at $t=0^+$ (immediately after a sudden change). An inductor opposes a sudden change in current. Therefore, the current through an inductor cannot change instantaneously, i.e., $i_L(0^+) = i_L(0^-)$. If the inductor was unenergized before $t=0$, its current at $t=0^+$ will be zero. In such a scenario, the inductor acts like an open circuit initially ($t=0^+$) in terms of current, but it can support a voltage across it. At $t=0^+$, if the current is zero, any voltage drop across the resistor ($iR$) will also be zero, and thus the entire source voltage appears across the inductor.

Answer 1

The Correct Option is C

The question asks for the voltage across the inductor at the instant the switch is closed (t=0).

1. Understanding the Concepts:

- Inductor Behavior at t=0: At the instant a switch is closed in a circuit with an inductor, the inductor acts as an open circuit. This is because the current through an inductor cannot change instantaneously. It opposes any sudden change in current.
- Initial Conditions: Before the switch closes (t < 0), the circuit is open, so no current flows through the inductor. At t = 0+, the inductor initially resists current flow.

2. Analyzing the Circuit:

When the switch is closed, the inductor initially acts as an open circuit. This means that at t = 0+, no current flows through the resistor. Therefore, there is no voltage drop across the resistor.

3. Applying Kirchhoff's Voltage Law (KVL):

According to KVL, the sum of the voltages around a closed loop must be zero. In this case, the loop consists of the voltage source, the resistor, and the inductor:
\( V_{source} - V_{resistor} - V_{inductor} = 0 \)

Since the current is initially zero, the voltage drop across the resistor is zero:
\( V_{resistor} = I \times R = 0 \times 30 = 0 \text{ V} \)

Therefore:
\( 60 - 0 - V_{inductor} = 0 \)
\( V_{inductor} = 60 \text{ V} \)

Final Answer:

The voltage across the inductor at t=0 is 60 V.