A series L-C-R circuit contains inductancle 5 mH, capacitor `2muF` and resistance `10 Omega`. If a frequency AC source is varied, then what is the frequency at which maximum power is dissipated?

Options

A. `(10^(5))/(pi)` Hz
B. `(10^(5))/(pi)`Hz
C. `2/3 xx 10^(5)` Hz
D. `5/pi xx 10^(3)` Hz

Explanation

To find the frequency at which maximum power is dissipated in a series L-C-R circuit, we need to calculate the resonant frequency. The resonant frequency \( f \) for a series L-C-R circuit is given by the formula:

\[ f = \frac{1}{2\pi\sqrt{LC}} \]

Where:
- \( L \) is the inductance in henries (H),
- \( C \) is the capacitance in farads (F).

### Step 1: Convert the given values to standard units
- Inductance \( L = 5 \, \text{mH} = 5 \times 10^{-3} \, \text{H} \)
- Capacitance \( C = 2 \, \mu\text{F} = 2 \times 10^{-6} \, \text{F} \)

### Step 2: Substitute the values into the resonant frequency formula
Now, substitute the values of \( L \) and \( C \) into the formula:

\[ f = \frac{1}{2\pi\sqrt{(5 \times 10^{-3})(2 \times 10^{-6})}} \]

### Step 3: Calculate the product \( LC \)
First, calculate \( LC \):

\[ LC = (5 \times 10^{-3})(2 \times 10^{-6}) = 10 \times 10^{-9} = 10^{-8} \, \text{H}\cdot\text{F} \]

### Step 4: Calculate the square root of \( LC \)
Now, find the square root:

\[ \sqrt{LC} = \sqrt{10^{-8}} = 10^{-4} \]

### Step 5: Substitute back into the frequency formula
Now substitute \( \sqrt{LC} \) back into the frequency formula:

\[ f = \frac{1}{2\pi (10^{-4})} \]

### Step 6: Simplify the expression
This simplifies to:

\[ f = \frac{10^4}{2\pi} \]

### Step 7: Calculate the final value
Now we can calculate the numerical value:

\[ f = \frac{10^4}{2 \times 3.14} \approx \frac{10^4}{6.28} \approx 1591.55 \, \text{Hz} \]

### Step 8: Express in terms of \( \frac{5}{\pi} \times 10^3 \)
To express it in the required format:

\[ f \approx \frac{5 \times 10^3}{\pi} \, \text{Hz} \]

Thus, the frequency at which maximum power is dissipated is:

\[ f = \frac{5}{\pi} \times 10^3 \, \text{Hz} \]

### Conclusion
The correct option is \( \text{D} \, \frac{5}{\pi} \times 10^3 \, \text{Hz} \).

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