A flat, circular coil of radius `10 cm` has 100 turns of wire. Auniform magnetic field exists in a direction perpendicular to the plane of the coil. This ficld is increasing at the rate of `0.1 T / s`. Calculate the magnitude of emf induced (in volt) in the coil.

Answer

To solve the problem, we will use Faraday's law of electromagnetic induction, which states that the induced electromotive force (emf) in a coil is equal to the negative rate of change of magnetic flux through the coil. The formula for induced emf (ε) is given by:

\[ \epsilon = -N \frac{d\Phi}{dt} \]

Where:
- \(N\) = number of turns in the coil
- \(\frac{d\Phi}{dt}\) = rate of change of magnetic flux

### Step 1: Calculate the area of the coil
The area \(A\) of a circular coil is given by the formula:

\[ A = \pi r^2 \]

Where \(r\) is the radius of the coil. Given that the radius \(r = 10 \, \text{cm} = 0.1 \, \text{m}\):

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

### Step 2: Calculate the rate of change of magnetic flux
The magnetic flux \(\Phi\) through the coil is given by:

\[ \Phi = B \cdot A \]

Where \(B\) is the magnetic field. The rate of change of magnetic flux is:

\[ \frac{d\Phi}{dt} = A \frac{dB}{dt} \]

Given that \(\frac{dB}{dt} = 0.1 \, \text{T/s}\):

\[ \frac{d\Phi}{dt} = A \cdot \frac{dB}{dt} = (0.01\pi) \cdot (0.1) = 0.001\pi \, \text{Wb/s} \]

### Step 3: Calculate the induced emf
Now, substituting the values into the induced emf formula:

\[ \epsilon = -N \frac{d\Phi}{dt} \]

Given that \(N = 100\):

\[ \epsilon = -100 \cdot (0.001\pi) = -0.1\pi \, \text{V} \]

### Step 4: Calculate the numerical value
Using the value of \(\pi \approx 3.14\):

\[ \epsilon \approx -0.1 \cdot 3.14 = -0.314 \, \text{V} \]

Thus, the magnitude of the induced emf is:

\[ \epsilon \approx 0.314 \, \text{V} \]

### Final Answer:
The magnitude of the induced emf in the coil is approximately **0.314 V**.

---