A circular coil of radius R = 10cm having 500 turns and total resistance `2Omega` is placed initially perpendicular to the earth's magnetic field of `3xx10^(-5)`T. The coil is rotated about its vertical diameter by an angle `pi` in 0.5 second. The induced current in the coil is

Options

A. 0.5 mA
B. 1.0 mA
C. 1.5 mA
D. 3.0 mA

Explanation

To solve the problem step by step, we will follow these procedures:

### Step 1: Calculate the Area of the Coil
The area \( A \) of a circular coil can be calculated using the formula:
\[ A = \pi R^2 \]
where \( R \) is the radius of the coil.

Given:
- Radius \( R = 10 \, \text{cm} = 0.1 \, \text{m} \)

Calculating the area:
\[ A = \pi (0.1)^2 = \pi (0.01) = 0.01\pi \, \text{m}^2 \]

### Step 2: Calculate the Change in Magnetic Flux
The magnetic flux \( \Phi \) through the coil is given by:
\[ \Phi = B \cdot A \cdot \cos(\theta) \]
where \( B \) is the magnetic field, \( A \) is the area, and \( \theta \) is the angle between the magnetic field and the normal to the coil's surface.

Initially, the coil is perpendicular to the magnetic field, so:
- Initial angle \( \theta_1 = 0^\circ \) (or \( 0 \, \text{rad} \))
- Final angle \( \theta_2 = 180^\circ \) (or \( \pi \, \text{rad} \))

Calculating the initial and final magnetic flux:
\[ \Phi_1 = B \cdot A \cdot \cos(0) = B \cdot A \]
\[ \Phi_2 = B \cdot A \cdot \cos(\pi) = -B \cdot A \]

Thus, the change in flux \( \Delta \Phi \) is:
\[ \Delta \Phi = \Phi_2 - \Phi_1 = -BA - BA = -2BA \]

Substituting the values:
\[ B = 3 \times 10^{-5} \, \text{T}, \quad A = 0.01\pi \, \text{m}^2 \]
\[ \Delta \Phi = -2 \cdot (3 \times 10^{-5}) \cdot (0.01\pi) = -6 \times 10^{-7}\pi \, \text{Wb} \]

### Step 3: Calculate the Induced EMF
According to Faraday's law of electromagnetic induction, the induced EMF \( \mathcal{E} \) is given by:
\[ \mathcal{E} = -\frac{\Delta \Phi}{\Delta t} \]
where \( \Delta t \) is the time interval during which the change occurs.

Given \( \Delta t = 0.5 \, \text{s} \):
\[ \mathcal{E} = -\frac{-6 \times 10^{-7}\pi}{0.5} = \frac{6 \times 10^{-7}\pi}{0.5} = 12 \times 10^{-7}\pi \, \text{V} \]
Calculating the numerical value:
\[ \mathcal{E} \approx 12 \times 10^{-7} \times 3.14 \approx 3.77 \times 10^{-6} \, \text{V} \approx 3.77 \, \text{mV} \]

### Step 4: Calculate the Induced Current
Using Ohm's law, the induced current \( I \) can be calculated as:
\[ I = \frac{\mathcal{E}}{R} \]
where \( R \) is the resistance of the coil.

Given \( R = 2 \, \Omega \):
\[ I = \frac{3.77 \times 10^{-6}}{2} \approx 1.885 \times 10^{-6} \, \text{A} \approx 1.89 \, \text{mA} \]

### Final Answer
The induced current in the coil is approximately \( 1.89 \, \text{mA} \).

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