A reacangular loop if size `(2m xx 1m)` is placed in x-y plane. A uniform but time varying magnetic field of strength T where t is the time elsapsed in second exists in sosace. The magnitude of induced emf (in V) at time t is

Options

A. 20=20i
B. 20
C. 20i
D. zero

Answer

To solve the problem of finding the induced EMF in a rectangular loop placed in a time-varying magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the dimensions of the loop**:
The rectangular loop has dimensions of 2 meters by 1 meter. Therefore, the area \( A \) of the loop can be calculated as:
\[ A = \text{length} \times \text{width} = 2 \, \text{m} \times 1 \, \text{m} = 2 \, \text{m}^2 \]

**Hint**: Remember that the area of a rectangle is simply the product of its length and width.

2. **Determine the magnetic field**:
The magnetic field \( \mathbf{B} \) is given as:
\[ \mathbf{B} = 2 \, \text{T} \, \hat{i} + 10 \, \text{T}^2 \, \hat{j} + 50 \, \text{T} \, \hat{k} \]
Here, the magnetic field has components in the x, y, and z directions.

**Hint**: Break down the magnetic field into its components to understand how it varies with time.

3. **Calculate the magnetic flux (\( \Phi \))**:
The magnetic flux through the loop is given by the dot product of the magnetic field and the area vector. Since the loop is in the x-y plane, the area vector \( \mathbf{A} \) points in the z-direction:
\[ \mathbf{A} = A \hat{k} = 2 \, \hat{k} \]
The magnetic flux \( \Phi \) is calculated as:
\[ \Phi = \mathbf{B} \cdot \mathbf{A} = (2 \hat{i} + 10 \hat{j} + 50 \hat{k}) \cdot (2 \hat{k}) = 50 \times 2 = 100 \, \text{Wb} \]
The contributions from the \( \hat{i} \) and \( \hat{j} \) components are zero because they are perpendicular to \( \hat{k} \).

**Hint**: Remember that the dot product only contributes when the vectors are in the same direction.

4. **Determine the induced EMF (\( \mathcal{E} \))**:
The induced EMF is given by Faraday's law of electromagnetic induction:
\[ \mathcal{E} = -\frac{d\Phi}{dt} \]
Since the magnetic flux \( \Phi \) is constant (as it does not depend on time), the derivative \( \frac{d\Phi}{dt} \) is zero:
\[ \mathcal{E} = -0 = 0 \, \text{V} \]

**Hint**: If the magnetic flux is constant, the induced EMF will be zero.

5. **Conclusion**:
The magnitude of the induced EMF at time \( t \) is:
\[ \mathcal{E} = 0 \, \text{V} \]

### Final Answer:
The magnitude of induced EMF at time \( t \) is \( 0 \, \text{V} \).