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 solve the problem, we need to find the r.m.s voltage across the capacitor (V_C) in a series R-C circuit given the r.m.s voltage of the source (V_S) and the r.m.s voltage across the resistor (V_R).

### Step-by-Step Solution:

1. **Identify the given values:**
- The r.m.s voltage of the AC source (V_S) = 50 V
- The r.m.s voltage across the resistor (V_R) = 40 V

2. **Use the formula for the total voltage in an R-C circuit:**
In a series R-C circuit, the relationship between the source voltage, the voltage across the resistor, and the voltage across the capacitor is given by:
\[ V_S^2 = V_R^2 + V_C^2 \]
where:
- \( V_S \) = r.m.s voltage of the source
- \( V_R \) = r.m.s voltage across the resistor
- \( V_C \) = r.m.s voltage across the capacitor

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

4. **Calculate the squares:**
\[ 2500 = 1600 + V_C^2 \]

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 \( V_C \):**
\[ V_C = \sqrt{900} = 30 \text{ V} \]

### Final Answer:
The r.m.s voltage across the capacitor is **30 V**.

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