A rectangular copper coil is placed in a uniform magnetic field of induction 40 mT with its plane perpendicular to the field. The area of the coil is shrinking at a constant rate of `0.5m^(2)s^(-1)`. The emf induced in the coil is

Options

A. 10 mV
B. 20 mV
C. 80 mV
D. 40 mV

Explanation

To solve the problem, we need to calculate the induced electromotive force (emf) in a rectangular copper coil placed in a uniform magnetic field. The area of the coil is shrinking at a constant rate. Here’s how to approach the solution step by step:

### Step 1: Identify the given parameters
- Magnetic field induction (B) = 40 mT = 40 × 10^(-3) T = 0.04 T
- Rate of change of area (dA/dt) = -0.5 m²/s (negative because the area is shrinking)

### Step 2: Understand the relationship between magnetic flux and induced emf
The magnetic flux (Φ) through the coil is given by:
\[ \Phi = B \cdot A \cdot \cos(\theta) \]
Since the plane of the coil is perpendicular to the magnetic field, \(\theta = 0^\circ\) and \(\cos(0) = 1\). Therefore, the equation simplifies to:
\[ \Phi = B \cdot A \]

### Step 3: Differentiate the magnetic flux with respect to time
To find the induced emf (E), we use Faraday's law of electromagnetic induction, which states:
\[ E = -\frac{d\Phi}{dt} \]
Differentiating the magnetic flux with respect to time gives:
\[ \frac{d\Phi}{dt} = B \cdot \frac{dA}{dt} \]
Thus, the induced emf can be expressed as:
\[ E = -B \cdot \frac{dA}{dt} \]

### Step 4: Substitute the known values into the equation
Now, substituting the values we have:
\[ E = - (0.04 \, \text{T}) \cdot (-0.5 \, \text{m}^2/\text{s}) \]

### Step 5: Calculate the induced emf
Calculating the above expression:
\[ E = 0.04 \cdot 0.5 = 0.02 \, \text{V} \]
Converting this into millivolts:
\[ E = 20 \, \text{mV} \]

### Final Answer
The induced emf in the coil is **20 mV**.

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