The magnetic flux through a coil varies with time as `phi=5t^2-6t+9.` The ratio of E.M.F. at `t=0s" to "t=0.5s` will be

Options

A. (a)`9:1`
B. (b)`1:6`
C. (c)`6:1`
D. (d)`1:9`

Answer

To solve the problem, we need to find the ratio of the electromotive force (E.M.F.) at two different times, \( t = 0 \) seconds and \( t = 0.5 \) seconds, given the magnetic flux \( \phi(t) = 5t^2 - 6t + 9 \).

### Step-by-Step Solution:

1. **Identify the formula for E.M.F.**:
According to Faraday's law of electromagnetic induction, the E.M.F. (ε) induced in a coil is given by the negative rate of change of magnetic flux:
\[ \epsilon = -\frac{d\phi}{dt} \]

2. **Differentiate the flux function**:
We need to differentiate the given flux function \( \phi(t) = 5t^2 - 6t + 9 \) with respect to time \( t \):
\[ \frac{d\phi}{dt} = \frac{d}{dt}(5t^2 - 6t + 9) = 10t - 6 \]

3. **Calculate E.M.F. at \( t = 0 \) seconds**:
Substitute \( t = 0 \) into the differentiated equation:
\[ \epsilon(0) = -\frac{d\phi}{dt}\bigg|_{t=0} = -(10(0) - 6) = -(-6) = 6 \, \text{V} \]

4. **Calculate E.M.F. at \( t = 0.5 \) seconds**:
Substitute \( t = 0.5 \) into the differentiated equation:
\[ \epsilon(0.5) = -\frac{d\phi}{dt}\bigg|_{t=0.5} = -(10(0.5) - 6) = -(5 - 6) = -(-1) = 1 \, \text{V} \]

5. **Calculate the ratio of E.M.F.s**:
Now, we find the ratio of E.M.F. at \( t = 0 \) seconds to that at \( t = 0.5 \) seconds:
\[ \text{Ratio} = \frac{\epsilon(0)}{\epsilon(0.5)} = \frac{6}{1} = 6 \]

### Final Answer:
The ratio of E.M.F. at \( t = 0 \) seconds to \( t = 0.5 \) seconds is \( 6 \).