Resonant frequency is given by (for a series L-C-R circuit)

Options

A. `2 pi sqrt(L/C)`
B. `1/(2pi) sqrt(L/C)`
C. `(2pi)/(sqrt(LC))`
D. `1/(2pisqrt(LC))`

Explanation

To find the resonant frequency of a series L-C-R circuit, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Components**: In a series L-C-R circuit, we have three main components:
- **L**: Inductance (measured in Henrys)
- **C**: Capacitance (measured in Farads)
- **R**: Resistance (measured in Ohms)

2. **Identify Reactances**: At resonance, the inductive reactance (XL) and capacitive reactance (XC) are equal:
- Inductive Reactance: \( X_L = \omega L \)
- Capacitive Reactance: \( X_C = \frac{1}{\omega C} \)

3. **Set Reactances Equal**: At resonance, we set \( X_L \) equal to \( X_C \):
\[ \omega L = \frac{1}{\omega C} \]

4. **Rearrange the Equation**: Multiply both sides by \( \omega \) to eliminate the fraction:
\[ \omega^2 L = \frac{1}{C} \]

5. **Solve for Omega**: Rearranging gives us:
\[ \omega^2 = \frac{1}{LC} \]
Taking the square root of both sides results in:
\[ \omega = \frac{1}{\sqrt{LC}} \]

6. **Convert Omega to Frequency**: The relationship between angular frequency (\(\omega\)) and frequency (f) is given by:
\[ f = \frac{\omega}{2\pi} \]
Substituting for \(\omega\):
\[ f = \frac{1}{2\pi \sqrt{LC}} \]

7. **Final Expression**: The resonant frequency \( f \) of the series L-C-R circuit is:
\[ f = \frac{1}{2\pi \sqrt{LC}} \]