In a series R-L-C circuit at resonance, the voltages across the resistor and the inductors are 30V and 40V…

1. Analyze the Question

We are given a series R-L-C circuit operating at resonance. We know the voltage across the resistor ($V_R = 30\text{V}$) and the voltage across the inductor ($V_L = 40\text{V}$). We need to find the voltage across the capacitor ($V_C$).

2. Explain the Concept

In a series R-L-C circuit, resonance occurs when the inductive reactance ($X_L$) is equal to the capacitive reactance ($X_C$).

Mathematically:

Since the same current ($I$) flows through all components in a series circuit, we use Ohm's Law ($V = IR$) to relate the voltages:

At resonance, the voltage drop across the inductor is exactly equal in magnitude to the voltage drop across the capacitor. These two voltages are $180^\circ$ out of phase, which is why they cancel each other out, leaving the total supply voltage equal to the voltage across the resistor ($V_{total} = V_R$).

3. Show the Solution

• Given: • $V_R = 30\text{V}$ • $V_L = 40\text{V}$
• Condition: The circuit is at resonance.
• Principle: At resonance, $V_L = V_C$.
• Calculation: Since $V_L = 40\text{V}$ and $V_L = V_C$, then: $$V_C = 40\text{V}$$

4. Identify the Answer

The voltage across the capacitor must be 40V to satisfy the condition of resonance. This matches option B.

5. Exam Tip

Always remember the "Resonance Rule": At resonance, the inductive and capacitive effects cancel each other out. If you see a question about a series R-L-C circuit at resonance, immediately look for the equality between $V_L$ and $V_C$ or $X_L$ and $X_C$. Don't let the resistor voltage distract you—it is only there to tell you the total voltage of the circuit!