The magnetic flux linked with a coil is given by an equation `phi` (in webers ) `= 8t^(2)+3t+5` . The induced e.m.f. in the coil at the fourth second will be

Options

A. 16 units
B. 39 units
C. 67 units
D. 145 units

Answer

To solve the problem of finding the induced e.m.f. in the coil at the fourth second, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given magnetic flux equation**:
The magnetic flux linked with the coil is given by the equation:
\[ \phi(t) = 8t^2 + 3t + 5 \]

2. **Use the formula for induced e.m.f.**:
The induced e.m.f. (ε) in the coil is given by Faraday's law of electromagnetic induction:
\[ \epsilon = -\frac{d\phi}{dt} \]

3. **Differentiate the flux equation**:
We need to differentiate the flux equation with respect to time (t):
\[ \frac{d\phi}{dt} = \frac{d}{dt}(8t^2 + 3t + 5) \]
Using the power rule of differentiation:
\[ \frac{d\phi}{dt} = 16t + 3 \]

4. **Substitute the value of t**:
We need to find the induced e.m.f. at the fourth second, which means we need to evaluate it at \( t = 4 \) seconds:
\[ \frac{d\phi}{dt} \bigg|_{t=4} = 16(4) + 3 = 64 + 3 = 67 \]

5. **Calculate the induced e.m.f.**:
Now, substitute this value back into the induced e.m.f. formula:
\[ \epsilon = -\frac{d\phi}{dt} = -67 \text{ volts} \]

6. **Interpret the result**:
The negative sign indicates the direction of the induced e.m.f. according to Lenz's law, but the magnitude of the induced e.m.f. is 67 volts.

### Final Answer:
The induced e.m.f. in the coil at the fourth second is:
\[ \epsilon = -67 \text{ volts} \]