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 will follow these steps:

### Step 1: Understand the given information
We have a magnetic flux \(\phi\) given by the equation:
\[ \phi = xt^2 \]
where \(x\) is a constant, and \(t\) is time in seconds. We also know that at \(t = 3\) seconds, the induced electromotive force (emf) is \(9\) volts.

### Step 2: Apply Faraday's Law of Electromagnetic Induction
According to Faraday's law, the induced emf (\( \mathcal{E} \)) is equal to the negative rate of change of magnetic flux:
\[ \mathcal{E} = -\frac{d\phi}{dt} \]

### Step 3: Differentiate the flux with respect to time
We need to find \(\frac{d\phi}{dt}\):
\[ \phi = xt^2 \]
Differentiating \(\phi\) with respect to \(t\):
\[ \frac{d\phi}{dt} = \frac{d}{dt}(xt^2) = 2xt \]

### Step 4: Set up the equation for induced emf
Now, substituting the expression for \(\frac{d\phi}{dt}\) into the equation for emf:
\[ \mathcal{E} = -2xt \]
At \(t = 3\) seconds, we know \(\mathcal{E} = 9\) volts:
\[ 9 = -2x(3) \]

### Step 5: Solve for \(x\)
Now, we can solve for \(x\):
\[ 9 = -6x \]
Dividing both sides by \(-6\):
\[ x = -\frac{9}{6} = -\frac{3}{2} = -1.5 \]

### Final Answer
Thus, the value of \(x\) is:
\[ \boxed{-1.5} \]

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