A flate circular coil of `10 cm` radius has `200` turns of wire the coil is connected to a capacitor of `20 muF` placed in a uniform magnetic field whise induction decreases at a rate of `0.01 Ts^(-1)` Find the charge on capacitor

Options

A. `0.51 muC`
B. `0.75 muC`
C. `0.92 muC`
D. `1.25 muC`

Explanation

To solve the problem step by step, we will follow the principles of electromagnetic induction and the relationship between charge, capacitance, and induced EMF.

### Step 1: Understand the given data
- Radius of the coil (r) = 10 cm = 0.1 m (convert cm to m)
- Number of turns (N) = 200
- Capacitance of the capacitor (C) = 20 µF = 20 × 10^(-6) F
- Rate of change of magnetic induction (dB/dt) = -0.01 T/s (negative sign indicates a decrease)

### Step 2: Calculate the area of the coil
The area (A) of the circular coil can be calculated using the formula:
\[ A = \pi r^2 \]
Substituting the radius:
\[ A = \pi (0.1)^2 = \pi \times 0.01 \, \text{m}^2 \]
\[ A \approx 0.0314 \, \text{m}^2 \]

### Step 3: Calculate the induced EMF (E)
The induced EMF in the coil can be calculated using Faraday's law of electromagnetic induction:
\[ E = -N \frac{d\Phi}{dt} \]
Where \(\Phi\) is the magnetic flux given by:
\[ \Phi = B \cdot A \]
Since the magnetic field is changing, we can express the change in flux as:
\[ \frac{d\Phi}{dt} = A \frac{dB}{dt} \]
Substituting the values:
\[ \frac{d\Phi}{dt} = A \cdot \left(-0.01\right) \]
\[ \frac{d\Phi}{dt} = 0.0314 \cdot (-0.01) \approx -3.14 \times 10^{-4} \, \text{Wb/s} \]

Now substituting this into the EMF equation:
\[ E = -200 \cdot (-3.14 \times 10^{-4}) \]
\[ E \approx 0.0628 \, \text{V} \]

### Step 4: Calculate the charge on the capacitor (Q)
Using the relationship between charge (Q), capacitance (C), and voltage (E):
\[ Q = C \cdot E \]
Substituting the values:
\[ Q = 20 \times 10^{-6} \cdot 0.0628 \]
\[ Q \approx 1.256 \times 10^{-6} \, \text{C} \]
\[ Q \approx 1.25 \, \mu C \]

### Final Answer
The charge on the capacitor is approximately \( 1.25 \, \mu C \).

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