An ac source of 50 V (r.m.s value) is connected across a series R - C circuit. If the r.m.s voltage across the resistor is 40 V, then the r.m.s voltage across the capacitor is

Options

A. 10 V
B. 20 V
C. 30 V
D. 40 V

Answer

To find the r.m.s voltage across the capacitor in the given series R-C circuit, we can use the relationship between the total voltage, the voltage across the resistor, and the voltage across the capacitor.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Total r.m.s voltage (V) = 50 V
- r.m.s voltage across the resistor (V_R) = 40 V

2. **Use the Voltage Relationship in a Series Circuit:**
In a series R-C circuit, the total voltage (V) is related to the voltages across the resistor (V_R) and the capacitor (V_C) by the following equation:
\[ V^2 = V_R^2 + V_C^2 \]

3. **Substitute the Known Values:**
Substitute the known values into the equation:
\[ 50^2 = 40^2 + V_C^2 \]

4. **Calculate the Squares:**
Calculate \(50^2\) and \(40^2\):
\[ 2500 = 1600 + V_C^2 \]

5. **Rearrange the Equation to Solve for \(V_C^2\):**
\[ V_C^2 = 2500 - 1600 \]

6. **Perform the Subtraction:**
\[ V_C^2 = 900 \]

7. **Take the Square Root to Find \(V_C\):**
\[ V_C = \sqrt{900} = 30 \text{ V} \]

### Conclusion:
The r.m.s voltage across the capacitor (V_C) is **30 V**.