A rectangular loop of sides `a` and `b` is placed in `xy`-placed. A uniform but time varying magnetic field of strength `B=20thati+10t^2hatj+50hatk` is present in the region. The magnitude of induced emf in the loop at time is

Options

A. `20+20t`
B. `20`
C. `20t`
D. zero

Explanation

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

### Step 1: Define the Area Vector
The rectangular loop lies in the xy-plane. Therefore, the area vector \( \vec{A} \) of the loop can be defined as:
\[ \vec{A} = A \hat{k} = ab \hat{k} \]
where \( a \) and \( b \) are the lengths of the sides of the rectangle, and \( \hat{k} \) indicates that the area vector is perpendicular to the plane of the loop.

### Step 2: Define the Magnetic Field
The magnetic field \( \vec{B} \) is given as:
\[ \vec{B} = 20t \hat{i} + 10t^2 \hat{j} + 50 \hat{k} \]

### Step 3: Calculate the Magnetic Flux
The magnetic flux \( \Phi \) through the loop is given by the dot product of the magnetic field \( \vec{B} \) and the area vector \( \vec{A} \):
\[ \Phi = \vec{B} \cdot \vec{A} = (20t \hat{i} + 10t^2 \hat{j} + 50 \hat{k}) \cdot (ab \hat{k}) \]
Calculating the dot product:
\[ \Phi = 20t \cdot 0 + 10t^2 \cdot 0 + 50 \cdot ab = 50ab \]

### Step 4: Determine the Induced EMF
The induced emf \( \mathcal{E} \) in the loop can be calculated using Faraday's law of electromagnetic induction, which states:
\[ \mathcal{E} = -\frac{d\Phi}{dt} \]
Since the magnetic flux \( \Phi \) is constant (it does not depend on time), the derivative of the flux with respect to time is:
\[ \frac{d\Phi}{dt} = 0 \]
Thus, the induced emf is:
\[ \mathcal{E} = -0 = 0 \]

### Conclusion
The magnitude of the induced emf in the loop at any time \( t \) is:
\[ \mathcal{E} = 0 \]

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