A time varying magnetic flux passing through a coil is given by `phi=xt^(2)`, if at `t=3s`, the emf induced is 9 V, then the value of x is

Options

A. `0.66Wb//s^(2)`
B. `-0.66Wb//s^(2)`
C. `1.5Wb//s^(2)`
D. `-1.5Wb//s^(2)`

Answer

To solve the problem, we need to find the value of \( x \) given the time-varying magnetic flux \( \phi = x t^2 \) and the induced emf at \( t = 3 \) seconds is 9 V.

### Step-by-Step Solution:

1. **Understand the formula for induced emf**:
The induced emf (\( \mathcal{E} \)) in a coil is given by Faraday's law of electromagnetic induction:
\[ \mathcal{E} = -\frac{d\phi}{dt} \]
where \( \phi \) is the magnetic flux.

2. **Differentiate the magnetic flux**:
Given \( \phi = x t^2 \), we need to differentiate this with respect to time \( t \):
\[ \frac{d\phi}{dt} = \frac{d}{dt}(x t^2) = x \cdot \frac{d}{dt}(t^2) = x \cdot 2t \]
Therefore,
\[ \frac{d\phi}{dt} = 2xt \]

3. **Substitute into the emf formula**:
Now, substituting \( \frac{d\phi}{dt} \) into the induced emf formula:
\[ \mathcal{E} = -\frac{d\phi}{dt} = -2xt \]

4. **Set up the equation using the given emf**:
We know that at \( t = 3 \) seconds, the induced emf \( \mathcal{E} \) is 9 V. Thus, we can write:
\[ 9 = -2x(3) \]

5. **Solve for \( x \)**:
Rearranging the equation gives:
\[ 9 = -6x \]
Dividing both sides by -6:
\[ x = -\frac{9}{6} = -1.5 \]

### Final Answer:
The value of \( x \) is \( -1.5 \, \text{Wb/s}^2 \).

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