In an RLC circuit, capacitance is changed from C to 2C. For the resonant frequency to remain unchanged, the inductance should be changed from L to :

Options

A. 4L
B. 2L
C. L/2
D. L/4

Answer

To solve the problem step by step, we need to understand the relationship between the resonant frequency, capacitance, and inductance in an RLC circuit.

### Step 1: Understand the formula for resonant frequency
The resonant frequency (ω) of an RLC circuit is given by the formula:
\[ \omega = \frac{1}{\sqrt{LC}} \]
where:
- \(L\) is the inductance,
- \(C\) is the capacitance.

### Step 2: Set up the equation for initial conditions
Let the initial capacitance be \(C_1 = C\) and the initial inductance be \(L_1 = L\). The initial resonant frequency is:
\[ \omega_1 = \frac{1}{\sqrt{L_1 C_1}} = \frac{1}{\sqrt{LC}} \]

### Step 3: Set up the equation for new conditions
When the capacitance is changed to \(C_2 = 2C\), we need to find the new inductance \(L_2\) such that the resonant frequency remains unchanged:
\[ \omega_2 = \frac{1}{\sqrt{L_2 C_2}} = \frac{1}{\sqrt{L_2 \cdot 2C}} \]

### Step 4: Equate the two frequencies
Since we want the resonant frequency to remain unchanged, we set \(\omega_1 = \omega_2\):
\[ \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{L_2 \cdot 2C}} \]

### Step 5: Cross-multiply to eliminate the fractions
Cross-multiplying gives:
\[ \sqrt{L_2 \cdot 2C} = \sqrt{LC} \]

### Step 6: Square both sides to eliminate the square root
Squaring both sides results in:
\[ L_2 \cdot 2C = LC \]

### Step 7: Solve for \(L_2\)
Now, we can solve for \(L_2\):
\[ L_2 \cdot 2C = LC \implies L_2 = \frac{LC}{2C} = \frac{L}{2} \]

### Final Answer
Thus, the inductance should be changed from \(L\) to:
\[ L_2 = \frac{L}{2} \]

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