if current in solenoid is `i_(1)=alphate^(betat)`. Which of the followign is correct graph between induced current rod time `(alpha` and `beta` are positive)

Options

A.
B.
C.
D.

Answer

To solve the problem, we need to find the induced current in the solenoid based on the given current \( i_1 = \alpha t e^{\beta t} \). The induced current is given by the rate of change of the current with respect to time, which is mathematically represented as:

\[ I_{\text{induced}} = \frac{dI_1}{dt} \]

### Step 1: Differentiate the current \( i_1 \)

Given:
\[ I_1 = \alpha t e^{\beta t} \]

We will use the product rule for differentiation, which states that if you have a product of two functions \( u(t) \) and \( v(t) \), then:

\[ \frac{d(uv)}{dt} = u \frac{dv}{dt} + v \frac{du}{dt} \]

Here, let:
- \( u = \alpha t \)
- \( v = e^{\beta t} \)

Now, we differentiate \( u \) and \( v \):

\[ \frac{du}{dt} = \alpha \quad \text{and} \quad \frac{dv}{dt} = \beta e^{\beta t} \]

Using the product rule:

\[ \frac{dI_1}{dt} = u \frac{dv}{dt} + v \frac{du}{dt} \]

Substituting the values:

\[ \frac{dI_1}{dt} = (\alpha t)(\beta e^{\beta t}) + (e^{\beta t})(\alpha) \]

### Step 2: Simplify the expression

Now, factor out the common terms:

\[ \frac{dI_1}{dt} = \alpha e^{\beta t} \left( t \beta + 1 \right) \]

### Step 3: Analyze the expression

Now we have:

\[ I_{\text{induced}} = \alpha e^{\beta t} (t \beta + 1) \]

### Step 4: Determine the sign of the induced current

1. **At \( t = 0 \)**:
\[ I_{\text{induced}} = \alpha e^{0} (0 \cdot \beta + 1) = \alpha \cdot 1 \cdot 1 = \alpha \]
Since \( \alpha > 0 \), \( I_{\text{induced}} > 0 \).

2. **As \( t \) increases**:
- The term \( e^{\beta t} \) is always positive for positive \( \beta \).
- The term \( (t \beta + 1) \) will also be positive for \( t > 0 \) since \( t \) is positive and \( \beta \) is positive.

Thus, \( I_{\text{induced}} \) is positive for all \( t \geq 0 \).

### Conclusion

The correct graph of the induced current with respect to time will start from a positive value at \( t = 0 \) and will continue to increase as \( t \) increases.