In an LCR circuit, the resonating frequency is 500 kHz. If the value of L is increased two times and value of C is decreased `(1)/(8)` times, then the new resonating frequency in kHz will be -

Options

A. 250
B. 500
C. 1000
D. 2000

Answer

To find the new resonating frequency of the LCR circuit after changing the values of inductance (L) and capacitance (C), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formula for resonating frequency**:
The resonating frequency \( f \) of an LCR circuit is given by the formula:
\[ f = \frac{1}{2\pi\sqrt{LC}} \]

2. **Identify the initial conditions**:
We are given that the initial resonating frequency \( f_1 \) is 500 kHz. We can express this in Hz for calculations:
\[ f_1 = 500 \times 10^3 \text{ Hz} \]

3. **Express the initial values of L and C**:
Let the initial values of inductance and capacitance be \( L_1 \) and \( C_1 \) respectively. Therefore, we can write:
\[ f_1 = \frac{1}{2\pi\sqrt{L_1C_1}} \]

4. **Change the values of L and C**:
According to the problem:
- The new inductance \( L_2 = 2L_1 \) (L is doubled)
- The new capacitance \( C_2 = \frac{1}{8}C_1 \) (C is decreased to one-eighth)

5. **Substitute the new values into the frequency formula**:
The new resonating frequency \( f_2 \) can be expressed as:
\[ f_2 = \frac{1}{2\pi\sqrt{L_2C_2}} = \frac{1}{2\pi\sqrt{(2L_1)(\frac{1}{8}C_1)}} \]

6. **Simplify the expression**:
\[ f_2 = \frac{1}{2\pi\sqrt{2L_1 \cdot \frac{1}{8}C_1}} = \frac{1}{2\pi\sqrt{\frac{2}{8}L_1C_1}} = \frac{1}{2\pi\sqrt{\frac{1}{4}L_1C_1}} = \frac{1}{2\pi\cdot\frac{1}{2}\sqrt{L_1C_1}} = \frac{2}{1}\cdot \frac{1}{2\pi\sqrt{L_1C_1}} \]
Therefore,
\[ f_2 = 2f_1 \]

7. **Calculate the new frequency**:
Since \( f_1 = 500 \text{ kHz} \):
\[ f_2 = 2 \times 500 \text{ kHz} = 1000 \text{ kHz} \]

### Final Answer:
The new resonating frequency \( f_2 \) is **1000 kHz**.

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