A loop of area `1m^2` is placed in a magnetic field `B=2T`, such that plane of the loop is parallel to the magnetic field. If the loop is rotated by `180^@`, the amount of net charge passing through any point of loop, it its resistance is `10Omega` is

Options

A. `0.4C`
B. `0.2C`
C. `0.8C`
D. `0 C`

Explanation

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

### Step 1: Understand the initial conditions
- Area of the loop, \( A = 1 \, m^2 \)
- Magnetic field strength, \( B = 2 \, T \)
- Resistance of the loop, \( R = 10 \, \Omega \)
- The loop is initially parallel to the magnetic field.

### Step 2: Calculate the initial magnetic flux
The magnetic flux \( \Phi \) is given by the formula:
\[ \Phi = B \cdot A \cdot \cos(\theta) \]
Where \( \theta \) is the angle between the magnetic field and the normal to the surface of the loop. Initially, since the plane of the loop is parallel to the magnetic field, \( \theta = 0^\circ \).

Thus, the initial flux \( \Phi_i \) is:
\[ \Phi_i = B \cdot A \cdot \cos(0^\circ) = 2 \, T \cdot 1 \, m^2 \cdot 1 = 2 \, Wb \]

### Step 3: Calculate the final magnetic flux after rotation
After rotating the loop by \( 180^\circ \), the angle \( \theta \) becomes \( 180^\circ \). Therefore, the final flux \( \Phi_f \) is:
\[ \Phi_f = B \cdot A \cdot \cos(180^\circ) = 2 \, T \cdot 1 \, m^2 \cdot (-1) = -2 \, Wb \]

### Step 4: Calculate the change in magnetic flux
The change in magnetic flux \( \Delta \Phi \) is given by:
\[ \Delta \Phi = \Phi_f - \Phi_i = -2 \, Wb - 2 \, Wb = -4 \, Wb \]
However, we are interested in the magnitude of the change:
\[ |\Delta \Phi| = 4 \, Wb \]

### Step 5: Calculate the induced EMF using Faraday's Law
According to Faraday's Law of Electromagnetic Induction, the induced EMF \( \mathcal{E} \) is equal to the rate of change of magnetic flux:
\[ \mathcal{E} = -\frac{\Delta \Phi}{\Delta t} \]
Since we are not given a specific time interval \( \Delta t \), we will use the total change in flux for the calculation of charge.

### Step 6: Calculate the induced current
Using Ohm's Law, the induced current \( I \) can be calculated as:
\[ I = \frac{\mathcal{E}}{R} \]
Substituting the induced EMF:
\[ I = \frac{|\Delta \Phi|}{R} = \frac{4 \, Wb}{10 \, \Omega} = 0.4 \, A \]

### Step 7: Calculate the total charge passing through the loop
The total charge \( Q \) that passes through the loop can be calculated using the formula:
\[ Q = I \cdot t \]
If we assume that the time \( t \) during which the flux changes is 1 second (for simplicity), then:
\[ Q = 0.4 \, A \cdot 1 \, s = 0.4 \, C \]

### Final Answer
The amount of net charge passing through any point of the loop is \( 0.4 \, C \).

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