A circuit area `0.01m^(2)` is kept inside a magnetic field which is normal to its plane. The magentic field changes form 2 tesla 1 tesla to in 1 millisecond. If the resistance of the circuit is `2omega`. The rate of heat evolved is

Options

A. `5J//s`
B. `50J//s`
C. `0.05J//s`
D. `0.5J//s`

Explanation

To solve the problem step by step, we will follow these calculations:

### Step 1: Calculate the Initial Magnetic Flux (Φ₁)
The initial magnetic flux (Φ₁) is given by the formula:
\[ \Phi_1 = B_1 \times A \]
Where:
- \( B_1 = 2 \, \text{T} \) (initial magnetic field)
- \( A = 0.01 \, \text{m}^2 \) (area of the circuit)

Calculating:
\[ \Phi_1 = 2 \, \text{T} \times 0.01 \, \text{m}^2 = 0.02 \, \text{Wb} \]

### Step 2: Calculate the Final Magnetic Flux (Φ₂)
The final magnetic flux (Φ₂) is given by:
\[ \Phi_2 = B_2 \times A \]
Where:
- \( B_2 = 1 \, \text{T} \) (final magnetic field)

Calculating:
\[ \Phi_2 = 1 \, \text{T} \times 0.01 \, \text{m}^2 = 0.01 \, \text{Wb} \]

### Step 3: Calculate the Change in Magnetic Flux (ΔΦ)
The change in magnetic flux (ΔΦ) is:
\[ \Delta \Phi = \Phi_2 - \Phi_1 \]
Calculating:
\[ \Delta \Phi = 0.01 \, \text{Wb} - 0.02 \, \text{Wb} = -0.01 \, \text{Wb} \]

### Step 4: Calculate the Induced EMF (ε)
The induced EMF (ε) can be calculated using Faraday's law of electromagnetic induction:
\[ \epsilon = -\frac{\Delta \Phi}{\Delta t} \]
Where:
- \( \Delta t = 1 \, \text{ms} = 1 \times 10^{-3} \, \text{s} \)

Calculating:
\[ \epsilon = -\frac{-0.01 \, \text{Wb}}{1 \times 10^{-3} \, \text{s}} = \frac{0.01}{0.001} = 10 \, \text{V} \]

### Step 5: Calculate the Current (I)
Using Ohm's law, the current (I) can be calculated as:
\[ I = \frac{V}{R} \]
Where:
- \( V = \epsilon = 10 \, \text{V} \)
- \( R = 2 \, \Omega \)

Calculating:
\[ I = \frac{10 \, \text{V}}{2 \, \Omega} = 5 \, \text{A} \]

### Step 6: Calculate the Rate of Heat Evolved (P)
The rate of heat evolved (P) in the circuit can be calculated using Joule's law:
\[ P = I^2 R \]

Calculating:
\[ P = (5 \, \text{A})^2 \times 2 \, \Omega = 25 \times 2 = 50 \, \text{J/s} \]

### Final Answer
The rate of heat evolved is **50 Joules per second**.

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