An external magnetic field is decreased to zero, due to which a current is induced in a circular wire loop of radius r and resistance R placed in the field. This current will not become zero at the instant when B stops changing

Options

A. At the instant when external magnetic field stops changing (t=0), the current in the loop is `i_(0)`. The current in the loop as a function of time for `t gt 0` is given by `i_(0)e^(-2Rt//mu_(0)pi)`.
B. For the same as in option (a), the current in the loop as a function of time `t=0` is given by `(mu_(0)iR)/(2 r)`.
C. The time in which current in loop decreases to `10^(-3) i_(0)` (from t=0) for `R=100 Omega` and `r=5 cm` is given by `(3 pi^(2)1n 10)/(10^(10))s`.
D. Fro the same as in option (c), the time in which current in loop decreases to `10^(-3) i_(0)` (from t=0) for `R=100 Omega` and `r-5 cm` is given by `(3pi^(2))/(10^(6))s`.

Explanation

To solve the problem, we need to analyze the situation where an external magnetic field is decreased to zero, causing an induced current in a circular wire loop. We will derive the expression for the induced current and evaluate the options provided.

### Step-by-Step Solution:

**Step 1: Understand the situation**
- We have a circular wire loop of radius \( r \) and resistance \( R \) placed in an external magnetic field \( B \).
- As the magnetic field decreases to zero, it induces a current in the loop according to Faraday's law of electromagnetic induction.

**Step 2: Apply Faraday's Law**
- According to Faraday's law, the induced electromotive force (emf) \( E \) in the loop is given by:
\[ E = -\frac{d\Phi}{dt} \]
where \( \Phi \) is the magnetic flux through the loop.

**Step 3: Calculate the magnetic flux**
- The magnetic flux \( \Phi \) through the loop is given by:
\[ \Phi = B \cdot A = B \cdot \pi r^2 \]
where \( A = \pi r^2 \) is the area of the loop.

**Step 4: Express the induced emf**
- If the magnetic field \( B \) is decreasing, we can express the change in flux as:
\[ E = -\frac{d}{dt}(B \cdot \pi r^2) \]
Since \( r \) is constant, we have:
\[ E = -\pi r^2 \frac{dB}{dt} \]

**Step 5: Relate induced current to emf**
- The induced current \( I \) in the loop can be related to the induced emf using Ohm's law:
\[ E = I \cdot R \]
Therefore:
\[ I = \frac{E}{R} = -\frac{\pi r^2}{R} \frac{dB}{dt} \]

**Step 6: Analyze the current at \( t = 0 \)**
- At the instant when \( B \) stops changing (let's denote this as \( t = 0 \)), the current \( I \) can be expressed as:
\[ I(t) = I_0 e^{-\frac{R t}{L}} \]
where \( L \) is the inductance of the loop.

**Step 7: Evaluate the options**
- **Option A** states that at \( t = 0 \), the current in the loop is \( I_0 \). This is correct because at the moment the magnetic field stops changing, the current is at its maximum value \( I_0 \).

- **Option B** is incorrect as it suggests a different relationship for the current at \( t = 0 \).

- **Option C** provides a specific time for the current to decrease to \( 10^{-3} I_0 \). We can calculate this using the derived equation.

- **Option D** is incorrect if it contradicts the findings in Option C.

### Conclusion:
The correct options are **A** and **C** based on the analysis of the induced current in the loop.