The flux linked with a circuit is given by `phi=t^(3)+3t-7` . The graph between time (`x`-axis) and induced emf (`y`-axis) will be

Options

A. straight line through origin
B. straight line with positive intercept
C. stairght line with negative intercept
D. parabola not through the origin

Explanation

To solve the problem, we need to find the induced electromotive force (emf) from the given magnetic flux function and then determine the nature of the graph between time (t) and induced emf (E).

### Step-by-Step Solution:

1. **Identify the given flux function**:
The magnetic flux linked with the circuit is given by:
\[ \phi(t) = t^3 + 3t - 7 \]

2. **Find the induced emf**:
The induced emf (E) is given by Faraday's law of electromagnetic induction, which states that:
\[ E = -\frac{d\phi}{dt} \]
We need to differentiate the flux function with respect to time (t).

3. **Differentiate the flux function**:
We will differentiate \(\phi(t)\):
\[ \frac{d\phi}{dt} = \frac{d}{dt}(t^3 + 3t - 7) \]
Using the power rule of differentiation:
\[ \frac{d\phi}{dt} = 3t^2 + 3 \]

4. **Calculate the induced emf**:
Now, substituting the derivative back into the equation for induced emf:
\[ E = -\frac{d\phi}{dt} = -(3t^2 + 3) = -3t^2 - 3 \]

5. **Analyze the expression for induced emf**:
The expression for the induced emf can be rewritten as:
\[ E = -3(t^2 + 1) \]
This shows that the induced emf is a quadratic function of time (t).

6. **Determine the nature of the graph**:
The expression \(-3(t^2 + 1)\) indicates that the graph is a downward-opening parabola because the coefficient of \(t^2\) is negative. Furthermore, since \(t^2 + 1\) is always positive for all real values of \(t\), the induced emf will always be negative.

7. **Check if the graph passes through the origin**:
To see if the graph passes through the origin, we check the value of induced emf when \(t = 0\):
\[ E(0) = -3(0^2 + 1) = -3 \]
Since \(E(0) \neq 0\), the graph does not pass through the origin.

### Conclusion:
The graph of induced emf (E) versus time (t) is a downward-opening parabola that does not pass through the origin.