In an alternating current circuit in which an inductance and capacitance are joined in series, current is found to be maximum when the value of inductance is 0.5 henry and the value of capacitance is 8µF. The angular frequency of applied alternating voltage will be

Options

A. 500
B. `5 xx 10^(4)`
C. 4000
D. 5000

Answer

To find the angular frequency of the applied alternating voltage in an LC circuit where the inductance (L) is 0.5 henry and the capacitance (C) is 8 µF, we can follow these steps:

### Step 1: Understand the resonance condition
In an LC circuit, the current is maximum at resonance when the inductive reactance (XL) equals the capacitive reactance (XC). This condition can be expressed as:
\[ X_L = X_C \]

### Step 2: Write the formulas for inductive and capacitive reactance
The inductive reactance (XL) and capacitive reactance (XC) are given by:
\[ X_L = \omega L \]
\[ X_C = \frac{1}{\omega C} \]
Where:
- \( \omega \) is the angular frequency,
- \( L \) is the inductance,
- \( C \) is the capacitance.

### Step 3: Set the reactances equal to each other
At resonance, we set \( X_L \) equal to \( X_C \):
\[ \omega L = \frac{1}{\omega C} \]

### Step 4: Rearrange the equation to find angular frequency
Multiplying both sides by \( \omega \) gives:
\[ \omega^2 = \frac{1}{LC} \]
Taking the square root of both sides, we find:
\[ \omega = \frac{1}{\sqrt{LC}} \]

### Step 5: Substitute the values of L and C
Now we can substitute the given values of inductance and capacitance into the equation. We have:
- \( L = 0.5 \, \text{H} \)
- \( C = 8 \, \mu\text{F} = 8 \times 10^{-6} \, \text{F} \)

Substituting these values:
\[ \omega = \frac{1}{\sqrt{0.5 \times 8 \times 10^{-6}}} \]

### Step 6: Calculate the value
Calculating the product:
\[ 0.5 \times 8 \times 10^{-6} = 4 \times 10^{-6} \]
Now, taking the square root:
\[ \sqrt{4 \times 10^{-6}} = 2 \times 10^{-3} \]
Thus:
\[ \omega = \frac{1}{2 \times 10^{-3}} = 500 \, \text{rad/s} \]

### Final Answer
The angular frequency of the applied alternating voltage is:
\[ \omega = 500 \, \text{rad/s} \]

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