To solve the problem of finding the voltage across the capacitor in a series R-L-C circuit at resonance, we need to understand the behavior of voltages in such a circuit. ### Step-by-Step Explanation 1. **Understanding Resonance in R-L-C Circuits**: - In a series R-L-C circuit, resonance occurs when the inductive reactance (XL) equals the capacitive reactance (XC). At this point, the circuit behaves as if it is purely resistive, and the total impedance is minimized. - At resonance, the current through the circuit is at its maximum, and the voltages across the individual components (resistor, inductor, and capacitor) can be different. 2. **Voltage Relationships**: - The voltage across the resistor (VR) is in phase with the current. - The voltage across the inductor (VL) leads the current by 90 degrees. - The voltage across the capacitor (VC) lags the current by 90 degrees. - At resonance, the voltages across the inductor and capacitor are equal in magnitude but opposite in phase, which means they cancel each other out when considering the total voltage in the circuit. 3. **Given Values**: - Voltage across the resistor (VR) = 30V - Voltage across the inductor (VL) = 40V 4. **Finding the Voltage Across the Capacitor (VC)**: - Since the inductor and capacitor voltages are equal in magnitude at resonance, we can express this relationship as: \[ VC = VL \] - Therefore, the voltage across the capacitor (VC) must also be 40V, but it is in the opposite phase to the inductor voltage. 5. **Total Voltage in the Circuit**: - The total voltage (V) in the circuit can be calculated using the Pythagorean theorem because the voltages across the resistor, inductor, and capacitor form a right triangle: \[ V = \sqrt{VR^2 + (VL - VC)^2} \] - Since \( VL = VC \), the equation simplifies to: \[ V = \sqrt{VR^2 + 0^2} = VR = 30V \] - This confirms that the voltage across the capacitor is indeed equal to the voltage across the inductor. ### Why the Other Options are Incorrect: - **Option A (30V)**: This is the voltage across the resistor, not the capacitor. The capacitor's voltage must be equal to the inductor's voltage at resonance. - **Option C (50V)**: This value does not correspond to any of the voltages in the circuit. It is not derived from the given values and does not follow the resonance condition. - **Option D (70V)**: This option suggests a total voltage that does not align with the calculated values. The maximum voltage in the circuit cannot exceed the sum of the voltages across the resistor and the inductor at resonance. ### Summary of Key Points: - At resonance in a series R-L-C circuit, the voltages across the inductor and capacitor are equal in magnitude but opposite in phase. - The voltage across the capacitor is equal to the voltage across the inductor, which is 40V in this case. - The voltage across the resistor is in phase with the current, while the voltages across the inductor and capacitor are out of phase. - The correct answer is **B. 40V**. This thorough understanding of resonance and voltage relationships in R-L-C circuits will help you tackle similar problems in your exams.