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

Explanation

To solve the problem, we need to find the ratio of the electromotive force (emf) at two different times, \( t = 3 \) seconds and \( t = 0 \) seconds, given the magnetic flux \( \phi(t) = 5t^2 + 6t + 9 \).

### Step-by-Step Solution:

1. **Understand the relationship between emf and magnetic flux**:
The induced emf (\( \varepsilon \)) in the coil is given by Faraday's law of electromagnetic induction:
\[ \varepsilon = -\frac{d\phi}{dt} \]
Here, \( \phi \) is the magnetic flux.

2. **Differentiate the magnetic flux with respect to time**:
Given \( \phi(t) = 5t^2 + 6t + 9 \), we differentiate this with respect to \( t \):
\[ \frac{d\phi}{dt} = \frac{d}{dt}(5t^2 + 6t + 9) \]
Using the power rule of differentiation:
\[ \frac{d\phi}{dt} = 10t + 6 \]

3. **Calculate the emf at \( t = 3 \) seconds**:
Substitute \( t = 3 \) into the expression for emf:
\[ \varepsilon(3) = 10(3) + 6 = 30 + 6 = 36 \, \text{V} \]

4. **Calculate the emf at \( t = 0 \) seconds**:
Substitute \( t = 0 \) into the expression for emf:
\[ \varepsilon(0) = 10(0) + 6 = 0 + 6 = 6 \, \text{V} \]

5. **Find the ratio of emf at \( t = 3 \) seconds to \( t = 0 \) seconds**:
The ratio of the emf at \( t = 3 \) seconds to the emf at \( t = 0 \) seconds is:
\[ \text{Ratio} = \frac{\varepsilon(3)}{\varepsilon(0)} = \frac{36}{6} = 6 \]

6. **Express the ratio in standard form**:
The ratio can be expressed as:
\[ \text{Ratio} = 6 : 1 \]

### Final Answer:
The ratio of emf at \( t = 3 \) seconds to \( t = 0 \) seconds is \( 6 : 1 \).