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 circuit consists of a voltage source, a switch, a resistor, and an inductor in series. 
The switch closes at $t=0$. We need to find the voltage across the inductor at $t=0$. 
For an inductor, the current cannot change instantaneously. 
Before the switch closes ($t<0$), there is no voltage source connected to the resistor-inductor series circuit. 
Thus, the current flowing through the inductor is zero: $i_L(0^-) = 0$ A. 
Since the current through an inductor cannot change instantaneously, the current just after the switch closes ($t=0^+$) will also be zero: $i_L(0^+) = i_L(0^-) = 0$ A. 
Now, let's apply Kirchhoff's Voltage Law (KVL) to the series circuit at $t=0^+$. The voltage source is $V_s = 60$ V. The resistor is $R = 30$ ohms. The inductor is $L = 15$ H. 
The KVL equation for the series circuit is: $V_s = V_R(t) + V_L(t)$ Where $V_R(t) = i(t)R$ is the voltage across the resistor and $V_L(t)$ is the voltage across the inductor. At $t=0^+$: $V_s = i(0^+)R + V_L(0^+)$ We know that $i(0^+) = 0$ A. Substitute the values into the KVL equation: $60 = (0) \times 30 + V_L(0^+)$ $60 = 0 + V_L(0^+)$ $V_L(0^+) = 60$ V. Therefore, the voltage across the inductor at $t=0$ (or more precisely, $t=0^+$) is 60 V.