A given wire is bent into a circular loop of radius `7cm` and is placed perpendicular to a magnetic field of `1.0T`.Within `0.1` second ,the loop is changed to a `100cm` square and the field increases to `1.8T` .The induced emf in the coil will be

Options

A. `13mV`
B. `26mV`
C. `39mV`
D. `52mV`

Answer

To solve the problem, we need to calculate the induced electromotive force (emf) in the wire when it is transformed from a circular loop to a square loop while the magnetic field changes. We will follow these steps:

### Step 1: Calculate the initial magnetic flux (Φ1)

The initial configuration is a circular loop with a radius of 7 cm (0.07 m) in a magnetic field of 1 T.

1. **Area of the circular loop (A1)**:
\[ A_1 = \pi r^2 = \pi (0.07)^2 = \pi (0.0049) \approx 0.0154 \, \text{m}^2 \]

2. **Magnetic flux (Φ1)**:
\[ \Phi_1 = B \cdot A_1 = 1 \, \text{T} \cdot 0.0154 \, \text{m}^2 \approx 0.0154 \, \text{Wb} \]

### Step 2: Calculate the final magnetic flux (Φ2)

The final configuration is a square loop with a side length of 100 cm (1 m) in a magnetic field of 1.8 T.

1. **Area of the square loop (A2)**:
\[ A_2 = \text{side}^2 = (1)^2 = 1 \, \text{m}^2 \]

2. **Magnetic flux (Φ2)**:
\[ \Phi_2 = B \cdot A_2 = 1.8 \, \text{T} \cdot 1 \, \text{m}^2 = 1.8 \, \text{Wb} \]

### Step 3: Calculate the change in magnetic flux (ΔΦ)

Now we can find the change in magnetic flux (ΔΦ):

\[ \Delta \Phi = \Phi_2 - \Phi_1 = 1.8 \, \text{Wb} - 0.0154 \, \text{Wb} \approx 1.7846 \, \text{Wb} \]

### Step 4: Calculate the induced emf (ε)

The induced emf can be calculated using Faraday's law of electromagnetic induction:

\[ \epsilon = -\frac{\Delta \Phi}{\Delta t} \]

Given that the time interval (Δt) is 0.1 seconds:

\[ \epsilon = -\frac{1.7846 \, \text{Wb}}{0.1 \, \text{s}} = -17.846 \, \text{V} \]

The negative sign indicates the direction of the induced emf according to Lenz's law, but we are interested in the magnitude:

\[ \epsilon \approx 17.85 \, \text{V} \]

### Final Answer

The induced emf in the coil will be approximately **17.85 V**.

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