The magnetic flux linked with a coil, in webers is given by the equation `phi=3t^(2)+4t+9`. Then, the magnitude of induced emf at t = 2 s

Options

A. 2 V
B. 4 V
C. 8 V
D. 16 V

Explanation

To find the magnitude of the induced electromotive force (emf) at \( t = 2 \) seconds, we will follow these steps:

### Step 1: Write down the equation for magnetic flux
The magnetic flux \( \phi \) linked with the coil is given by the equation:
\[ \phi(t) = 3t^2 + 4t + 9 \]

### Step 2: Differentiate the magnetic flux with respect to time
The induced emf (\( \mathcal{E} \)) is given by Faraday's law of electromagnetic induction, which states that the induced emf is equal to the negative rate of change of magnetic flux:
\[ \mathcal{E} = -\frac{d\phi}{dt} \]
Now, we will differentiate \( \phi(t) \):
\[ \frac{d\phi}{dt} = \frac{d}{dt}(3t^2 + 4t + 9) \]
Using the power rule of differentiation:
\[ \frac{d\phi}{dt} = 6t + 4 \]

### Step 3: Substitute \( t = 2 \) seconds into the derivative
Now we will substitute \( t = 2 \) seconds into the derivative to find the rate of change of flux at that moment:
\[ \frac{d\phi}{dt} \bigg|_{t=2} = 6(2) + 4 = 12 + 4 = 16 \]

### Step 4: Calculate the induced emf
Now, we can find the induced emf:
\[ \mathcal{E} = -\frac{d\phi}{dt} = -16 \]
Since we are interested in the magnitude of the induced emf, we take the absolute value:
\[ |\mathcal{E}| = 16 \text{ volts} \]

### Conclusion
The magnitude of the induced emf at \( t = 2 \) seconds is:
\[ \boxed{16 \text{ volts}} \]

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