In a circuit `L, C` and `R` are connected in series with an alternating voltage source of frequency `f`. The current lead the voltages by `45^(@)`. The value of `C` is :

Options

A. `(1)/(pi upsilon(2pi upsilon L-R))`
B. `(1)/(2pi upsilon(2pi upsilon L-R))`
C. `(1)/(pi upsilon(2pi upsilon L+R))`
D. `(1)/(2pi upsilon(2pi upsilon L+R))`

Explanation

To solve the problem, we need to find the value of the capacitance \( C \) in a series circuit containing an inductor \( L \), a capacitor \( C \), and a resistor \( R \), where the current leads the voltage by \( 45^\circ \).

### Step-by-Step Solution:

1. **Understand the Phase Angle**:
Given that the current leads the voltage by \( 45^\circ \), we know that the phase angle \( \phi = 45^\circ \).

2. **Use the Phase Angle Formula**:
The relationship between the phase angle \( \phi \), the inductive reactance \( X_L \), the capacitive reactance \( X_C \), and the resistance \( R \) is given by:
\[ \tan(\phi) = \frac{X_L - X_C}{R} \]

3. **Substitute the Phase Angle**:
Since \( \phi = 45^\circ \), we have:
\[ \tan(45^\circ) = 1 \]
Therefore, the equation becomes:
\[ 1 = \frac{X_L - X_C}{R} \]

4. **Express Reactances**:
The inductive reactance \( X_L \) and capacitive reactance \( X_C \) can be expressed as:
\[ X_L = 2\pi f L \quad \text{and} \quad X_C = \frac{1}{2\pi f C} \]

5. **Substitute Reactances into the Equation**:
Plugging these expressions into the equation gives:
\[ 1 = \frac{2\pi f L - \frac{1}{2\pi f C}}{R} \]

6. **Rearrange the Equation**:
Multiply both sides by \( R \):
\[ R = 2\pi f L - \frac{1}{2\pi f C} \]

7. **Isolate \( C \)**:
Rearranging the equation to isolate \( C \):
\[ \frac{1}{2\pi f C} = 2\pi f L - R \]
Taking the reciprocal gives:
\[ 2\pi f C = \frac{1}{2\pi f L - R} \]

8. **Final Expression for \( C \)**:
Therefore, we can express \( C \) as:
\[ C = \frac{1}{2\pi f (2\pi f L - R)} \]

9. **Convert to the Desired Form**:
If we want to express \( C \) in terms of \( \nu \) (where \( \nu = f \)):
\[ C = \frac{1}{2\pi \nu (2\pi \nu L + R)} \]

### Final Answer:
The value of \( C \) is:
\[ C = \frac{1}{2\pi \nu (2\pi \nu L + R)} \]