The reactance of a capacitor is `X_(1)` for frequency `n_(1)` and `X_(2)` for frequency `n_(2)` then `X_(1) : X_(2)` is

Options

A. `1:1`
B. `n_(1):n_(2)`
C. `n_(2):n_(1)`
D. `n_(1)^(2): n_(2)^(2)`

Answer

To solve the problem, we need to find the ratio of the reactance of a capacitor at two different frequencies. The reactance \( X \) of a capacitor is given by the formula:

\[ X_c = \frac{1}{2 \pi f C} \]

where:
- \( X_c \) is the capacitive reactance,
- \( f \) is the frequency,
- \( C \) is the capacitance.

### Step-by-Step Solution:

1. **Identify the Reactance at Different Frequencies**:
- For frequency \( n_1 \), the reactance \( X_1 \) is:
\[ X_1 = \frac{1}{2 \pi n_1 C} \]
- For frequency \( n_2 \), the reactance \( X_2 \) is:
\[ X_2 = \frac{1}{2 \pi n_2 C} \]

2. **Write the Ratio of Reactances**:
- We need to find the ratio \( \frac{X_1}{X_2} \):
\[ \frac{X_1}{X_2} = \frac{\frac{1}{2 \pi n_1 C}}{\frac{1}{2 \pi n_2 C}} \]

3. **Simplify the Ratio**:
- The \( 2 \pi \) and \( C \) terms in the numerator and denominator cancel out:
\[ \frac{X_1}{X_2} = \frac{1/n_1}{1/n_2} = \frac{n_2}{n_1} \]

4. **Final Result**:
- Therefore, the ratio of the reactance is:
\[ \frac{X_1}{X_2} = \frac{n_2}{n_1} \]

### Conclusion:
The ratio of the reactance of the capacitor at frequencies \( n_1 \) and \( n_2 \) is given by:
\[ X_1 : X_2 = n_2 : n_1 \]