A coil of inductance `5.0 mH` and negligible resistance is connected to an oscillator giving an output voltage `E=(10V) sin omegat`.Which of the following is correct

Options

A. for `omega=100 s^(-1)` peak current is `20 A`
B. for `omega=500 s^(-1)` peak current is `4 A`
C. for `omega=1000 s^(-1)` peak current is `2 A`
D. for `omega=1000 s^(-1)` peak current is `4 A`

Explanation

To solve the problem, we need to analyze the behavior of the coil when connected to an oscillator with a given output voltage. The coil has an inductance of \( L = 5.0 \, \text{mH} \) and negligible resistance. The output voltage from the oscillator is given by:

\[ E(t) = 10 \, \text{V} \cdot \sin(\omega t) \]

### Step 1: Identify the peak voltage
The peak voltage \( E_0 \) is given as \( 10 \, \text{V} \).

### Step 2: Use the formula for peak current in an inductor
The peak current \( I_0 \) in an inductor is given by the formula:

\[ I_0 = \frac{E_0}{\omega L} \]

### Step 3: Calculate for different values of \( \omega \)

#### Case 1: \( \omega = 100 \, \text{rad/s} \)
1. Substitute \( E_0 = 10 \, \text{V} \), \( \omega = 100 \, \text{rad/s} \), and \( L = 5.0 \, \text{mH} = 5 \times 10^{-3} \, \text{H} \):
\[ I_0 = \frac{10}{100 \times 5 \times 10^{-3}} = \frac{10}{0.5} = 20 \, \text{A} \]

#### Case 2: \( \omega = 500 \, \text{rad/s} \)
1. Substitute \( E_0 = 10 \, \text{V} \), \( \omega = 500 \, \text{rad/s} \):
\[ I_0 = \frac{10}{500 \times 5 \times 10^{-3}} = \frac{10}{2.5} = 4 \, \text{A} \]

#### Case 3: \( \omega = 1000 \, \text{rad/s} \)
1. Substitute \( E_0 = 10 \, \text{V} \), \( \omega = 1000 \, \text{rad/s} \):
\[ I_0 = \frac{10}{1000 \times 5 \times 10^{-3}} = \frac{10}{5} = 2 \, \text{A} \]

### Step 4: Analyze the options
- For \( \omega = 100 \, \text{rad/s} \), \( I_0 = 20 \, \text{A} \) (Correct)
- For \( \omega = 500 \, \text{rad/s} \), \( I_0 = 4 \, \text{A} \) (Correct)
- For \( \omega = 1000 \, \text{rad/s} \), \( I_0 = 2 \, \text{A} \) (Correct)
- The fourth option states that for \( \omega = 100 \, \text{rad/s} \), the peak value is \( 4 \, \text{A} \) which is incorrect.

### Conclusion
The correct options are:
- \( \omega = 100 \, \text{rad/s} \) gives \( 20 \, \text{A} \)
- \( \omega = 500 \, \text{rad/s} \) gives \( 4 \, \text{A} \)
- \( \omega = 1000 \, \text{rad/s} \) gives \( 2 \, \text{A} \)

Thus, the correct answers are the first three options.