A long solenoid of radius R carries a time (t)-dependent current `I(t)= I_(0)t^(2) (1-t)`. A conducting ring of radius 3R is placed co-axially near its middle. During the time interva `0 le t le 1`, the induced current `(I_(R ))` in the ring varies as: [Take resistance of ring to be `R_(0)`]

Options

A.
B.
C.
D.

Explanation

To solve the problem step by step, we will analyze the given information and apply relevant physics concepts.

### Step 1: Understand the Current in the Solenoid
The current in the solenoid is given by:
\[ I(t) = I_0 t^2 (1 - t) \]
This indicates that the current varies with time.

### Step 2: Calculate the Magnetic Field Inside the Solenoid
The magnetic field \( B \) inside a long solenoid is given by:
\[ B = \mu_0 n I \]
where \( n \) is the number of turns per unit length. Assuming the solenoid has \( n \) turns and using the given current:
\[ B(t) = \mu_0 n I_0 t^2 (1 - t) \]

### Step 3: Determine the Area of the Conducting Ring
The radius of the conducting ring is \( 3R \). The area \( A \) of the ring is:
\[ A = \pi (3R)^2 = 9\pi R^2 \]

### Step 4: Calculate the Magnetic Flux through the Ring
The magnetic flux \( \Phi \) through the ring is given by:
\[ \Phi = B \cdot A \]
Substituting the expressions for \( B \) and \( A \):
\[ \Phi(t) = \mu_0 n I_0 t^2 (1 - t) \cdot 9\pi R^2 \]
\[ \Phi(t) = 9\mu_0 n I_0 \pi R^2 t^2 (1 - t) \]

### Step 5: Find the Induced EMF
According to Faraday's law of electromagnetic induction, the induced EMF \( \mathcal{E} \) is given by:
\[ \mathcal{E} = -\frac{d\Phi}{dt} \]
Differentiating \( \Phi(t) \):
\[ \frac{d\Phi}{dt} = 9\mu_0 n I_0 \pi R^2 \frac{d}{dt}(t^2 (1 - t)) \]
Using the product rule:
\[ \frac{d}{dt}(t^2 (1 - t)) = 2t(1 - t) - t^2 = 2t - 3t^2 \]
Thus,
\[ \mathcal{E} = -9\mu_0 n I_0 \pi R^2 (2t - 3t^2) \]

### Step 6: Calculate the Induced Current
The induced current \( I_R \) in the ring can be calculated using Ohm's law:
\[ I_R = \frac{\mathcal{E}}{R_0} \]
Substituting for \( \mathcal{E} \):
\[ I_R = -\frac{9\mu_0 n I_0 \pi R^2 (2t - 3t^2)}{R_0} \]

### Step 7: Determine When the Induced Current is Zero
To find when the induced current is zero, set the expression \( 2t - 3t^2 = 0 \):
\[ t(2 - 3t) = 0 \]
This gives:
1. \( t = 0 \)
2. \( t = \frac{2}{3} \)

### Conclusion
The induced current varies as a function of time and is zero at \( t = 0 \) and \( t = \frac{2}{3} \). The nature of the function \( I_R \) indicates it is a quadratic function of time, which opens downwards (as indicated by the negative coefficient of \( t^2 \)).