The magnetic flux through a coil varies with time as `phi=5t^2+6t+9` The ratio of emf at t = 3 s to t = 0 s will be

Options

A. `9:1`
B. `1:6`
C. `6:1`
D. `1:9`

Answer

To solve the problem, we need to find the electromotive force (emf) at two different times (t = 3 seconds and t = 0 seconds) based on the given magnetic flux function, and then calculate the ratio of these emf values.

### Step-by-Step Solution:

1. **Given Magnetic Flux Function**:
The magnetic flux through the coil is given by:
\[ \phi(t) = 5t^2 + 6t + 9 \]

2. **Finding the emf**:
The electromotive force (emf) induced in the coil is related to the rate of change of magnetic flux through the coil. The formula for emf (ε) is:
\[ \text{emf} = -\frac{d\phi}{dt} \]

3. **Differentiate the Flux Function**:
We need to differentiate the magnetic flux function with respect to time (t):
\[ \frac{d\phi}{dt} = \frac{d}{dt}(5t^2 + 6t + 9) \]
Using the power rule of differentiation:
\[ \frac{d\phi}{dt} = 10t + 6 \]

4. **Calculate emf at t = 3 seconds**:
Substitute \( t = 3 \) into the derivative to find the emf:
\[ \text{emf at } t = 3 = -\frac{d\phi}{dt} \bigg|_{t=3} = -(10 \cdot 3 + 6) = -(30 + 6) = -36 \]

5. **Calculate emf at t = 0 seconds**:
Substitute \( t = 0 \) into the derivative to find the emf:
\[ \text{emf at } t = 0 = -\frac{d\phi}{dt} \bigg|_{t=0} = -(10 \cdot 0 + 6) = -6 \]

6. **Calculate the Ratio of emf**:
Now we find the ratio of emf at \( t = 3 \) seconds to emf at \( t = 0 \) seconds:
\[ \text{Ratio} = \frac{\text{emf at } t = 3}{\text{emf at } t = 0} = \frac{-36}{-6} = 6 \]

### Final Answer:
The ratio of emf at \( t = 3 \) seconds to \( t = 0 \) seconds is:
\[ \text{Ratio} = 6:1 \]