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

Explanation

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

### Step 1: Identify the given parameters
We have the following parameters:
- E.M.F. \( E(t) = 4 \cos(1000t) \) volts
- Inductance \( L = 3 \, \text{mH} = 3 \times 10^{-3} \, \text{H} \)
- Resistance \( R = 4 \, \Omega \)

### Step 2: Determine the angular frequency
The angular frequency \( \omega \) is given as the coefficient of \( t \) in the E.M.F. equation:
\[ \omega = 1000 \, \text{rad/s} \]

### Step 3: Calculate the inductive reactance
The inductive reactance \( X_L \) can be calculated using the formula:
\[ X_L = \omega L \]
Substituting the values:
\[ X_L = 1000 \times (3 \times 10^{-3}) = 3 \, \Omega \]

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

### Step 5: Calculate the amplitude of the current
The amplitude of the current \( I \) can be calculated using Ohm's law for AC circuits:
\[ I = \frac{E}{Z} \]
The amplitude of the voltage \( E \) is 4 volts, so:
\[ I = \frac{4}{5} = 0.8 \, \text{A} \]

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

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