An e.m.f. `E=4cos(1000t)`volt is applied to an `LR` circuit of inductance `3mH` and resistance `4ohm`. The amplitude of current in the circuit is

Options

A. `4sqrt7A`
B. `1.0A`
C. `(4)/(7)A`
D. `0.8A`

Answer

To find the amplitude of current in the given LR circuit, we can follow these steps:

### Step 1: Identify the given parameters
- The e.m.f. is given as \( E(t) = 4 \cos(1000t) \) volts.
- The inductance \( L = 3 \, \text{mH} = 3 \times 10^{-3} \, \text{H} \).
- The resistance \( R = 4 \, \Omega \).

### Step 2: Determine the angular frequency \( \omega \)
From the e.m.f. equation, the angular frequency \( \omega \) can be identified as:
\[ \omega = 1000 \, \text{rad/s} \]

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

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

### Step 5: Calculate the amplitude of current \( I_0 \)
The amplitude of the current \( I_0 \) is given by the formula:
\[ I_0 = \frac{E_0}{Z} \]
Where \( E_0 \) is the amplitude of the e.m.f. (which is 4 volts). Thus:
\[ I_0 = \frac{4 \, \text{V}}{5 \, \Omega} = 0.8 \, \text{A} \]

### Final Answer
The amplitude of current in the circuit is:
\[ \boxed{0.8 \, \text{A}} \]