In a LR circuit of 3 mH dinductance and `4 Omega` resistance, emf `E=4cos 1000t` volt is applied. The amplitude of current is

Options

A. `(4)/(sqrt(7))A`
B. `1.0A`
C. `(4)/(7)A`
D. `0.8A`

Explanation

To solve the problem step by step, we will follow the process of calculating the impedance of the LR circuit and then use it to find the amplitude of the current.

### Step 1: Identify given values
- Inductance \( L = 3 \, \text{mH} = 3 \times 10^{-3} \, \text{H} \)
- Resistance \( R = 4 \, \Omega \)
- EMF \( E(t) = 4 \cos(1000t) \)
- Angular frequency \( \omega = 1000 \, \text{rad/s} \)

### Step 2: Calculate the inductive reactance \( X_L \)
The inductive reactance \( X_L \) is given by the formula:
\[ X_L = \omega L \]
Substituting the values:
\[ X_L = 1000 \times (3 \times 10^{-3}) = 3 \, \Omega \]

### Step 3: Calculate the impedance \( Z \)
The impedance \( Z \) of an LR circuit is calculated using the formula:
\[ Z = \sqrt{R^2 + X_L^2} \]
Substituting the values:
\[ Z = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \, \Omega \]

### Step 4: Calculate the amplitude of the current \( I_0 \)
The amplitude of the current \( I_0 \) can be calculated using Ohm's law for AC circuits:
\[ I_0 = \frac{E_0}{Z} \]
Where \( E_0 \) is the amplitude of the EMF. Here, \( E_0 = 4 \, \text{V} \):
\[ I_0 = \frac{4}{5} = 0.8 \, \text{A} \]

### Final Answer
The amplitude of the current is \( 0.8 \, \text{A} \).

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