A circular coil of radius 8cm, 400 turns and resistance `2Omega` is placed with its plane perpendicular to the horizantal component of the earth's magnetic fiedl. It is rotated about its vertical diameter through `180^(@)` in 0.30 s. Horizontal component of earth magnitude of current induced in the coil is approximately

Options

A. `4xx10^(-2)A`
B. `8xx10^(-4)A`
C. `8xx10^(-2)A`
D. `1.92xx10^(-3)A`

Explanation

To find the magnitude of the current induced in the coil, we will follow these steps:

### Step 1: Calculate the Area of the Coil
The area \( A \) of a circular coil is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the coil. Given that the radius \( r = 8 \, \text{cm} = 0.08 \, \text{m} \):
\[ A = \pi (0.08)^2 = \pi (0.0064) \approx 0.0201 \, \text{m}^2 \]

### Step 2: Calculate the Initial Magnetic Flux
The initial magnetic flux \( \Phi_i \) when the coil is perpendicular to the magnetic field is given by:
\[ \Phi_i = B A \cos(0^\circ) = B A \]
Given that the horizontal component of the Earth's magnetic field \( B = 3 \times 10^{-5} \, \text{T} \):
\[ \Phi_i = 3 \times 10^{-5} \times 0.0201 \approx 6.03 \times 10^{-7} \, \text{Wb} \]

### Step 3: Calculate the Final Magnetic Flux
The final magnetic flux \( \Phi_f \) after rotating the coil through \( 180^\circ \) is given by:
\[ \Phi_f = B A \cos(180^\circ) = -B A \]
Thus:
\[ \Phi_f = -3 \times 10^{-5} \times 0.0201 \approx -6.03 \times 10^{-7} \, \text{Wb} \]

### Step 4: Calculate the Change in Magnetic Flux
The change in magnetic flux \( \Delta \Phi \) is:
\[ \Delta \Phi = \Phi_f - \Phi_i = -6.03 \times 10^{-7} - 6.03 \times 10^{-7} = -1.206 \times 10^{-6} \, \text{Wb} \]

### Step 5: Calculate the Induced EMF
According to Faraday's law of electromagnetic induction, the induced EMF \( E \) is given by:
\[ E = -N \frac{\Delta \Phi}{\Delta t} \]
where \( N \) is the number of turns and \( \Delta t \) is the time taken for the rotation (0.30 s). Given \( N = 400 \):
\[ E = -400 \frac{-1.206 \times 10^{-6}}{0.30} \approx 1.608 \times 10^{-3} \, \text{V} \approx 1.6 \, \text{mV} \]

### Step 6: Calculate the Induced Current
Using Ohm's law, the induced current \( I \) can be calculated as:
\[ I = \frac{E}{R} \]
where \( R \) is the resistance of the coil (2 Ω):
\[ I = \frac{1.608 \times 10^{-3}}{2} \approx 8.04 \times 10^{-4} \, \text{A} \approx 0.804 \, \text{mA} \]

### Final Answer
The magnitude of the current induced in the coil is approximately:
\[ I \approx 8 \times 10^{-4} \, \text{A} \]