A physicist works in a laboratory where the magnetic field is `2T`. She wears a necklace enclosing area `0.01 m^(2)` in such a way that the plane of the necklace is normal to the field and is having a resistance `R=0.01Omega`. Because of power failure, the field decays to `1T` in time `10^(-3)` seconds.
The what is the total heat produced in her necklace?`(T=tesla)`

Options

A. 10 J
B. 20 J
C. 30 J
D. 40 J

Explanation

To solve the problem step by step, we will follow the principles of electromagnetic induction and the relationship between induced electromotive force (emf), current, resistance, and heat produced.

### Step 1: Identify the parameters given in the problem
- Initial magnetic field, \( B_i = 2 \, \text{T} \)
- Final magnetic field, \( B_f = 1 \, \text{T} \)
- Area enclosed by the necklace, \( A = 0.01 \, \text{m}^2 \)
- Resistance of the necklace, \( R = 0.01 \, \Omega \)
- Time taken for the change in magnetic field, \( t = 10^{-3} \, \text{s} \)

### Step 2: Calculate the change in magnetic flux
The magnetic flux \( \Phi \) is given by:
\[ \Phi = B \cdot A \]
The change in magnetic flux \( \Delta \Phi \) when the magnetic field changes from \( B_i \) to \( B_f \) is:
\[ \Delta \Phi = \Phi_f - \Phi_i = (B_f \cdot A) - (B_i \cdot A) = A(B_f - B_i) \]
Substituting the values:
\[ \Delta \Phi = 0.01 \, \text{m}^2 \cdot (1 \, \text{T} - 2 \, \text{T}) = 0.01 \, \text{m}^2 \cdot (-1 \, \text{T}) = -0.01 \, \text{Wb} \]

### Step 3: Calculate the induced emf (E)
The induced emf \( E \) is given by Faraday's law of electromagnetic induction:
\[ E = -\frac{\Delta \Phi}{t} \]
Substituting the values:
\[ E = -\frac{-0.01 \, \text{Wb}}{10^{-3} \, \text{s}} = \frac{0.01}{10^{-3}} = 10 \, \text{V} \]

### Step 4: Calculate the current (I) using Ohm's law
Using Ohm's law, the current \( I \) can be calculated as:
\[ I = \frac{E}{R} \]
Substituting the values:
\[ I = \frac{10 \, \text{V}}{0.01 \, \Omega} = 1000 \, \text{A} \]

### Step 5: Calculate the heat produced (Q)
The heat produced in the necklace can be calculated using the formula:
\[ Q = I^2 R t \]
Substituting the values:
\[ Q = (1000 \, \text{A})^2 \cdot 0.01 \, \Omega \cdot 10^{-3} \, \text{s} \]
\[ Q = 1000000 \cdot 0.01 \cdot 10^{-3} = 10 \, \text{J} \]

### Final Answer
The total heat produced in her necklace is \( \boxed{10 \, \text{J}} \).

---