The flux linked with a coil at any instant 't' is given by `phi = 10t^(2) - 50t +250`
The induced emf at `t = 3s` is

Options

A. 10 V
B. 190 V
C. `-190 V`
D. `-10 V`

Answer

To find the induced electromotive force (emf) at \( t = 3 \) seconds, we will follow these steps:

### Step 1: Write down the expression for magnetic flux
The magnetic flux \( \Phi \) linked with the coil is given by:
\[ \Phi(t) = 10t^2 - 50t + 250 \]

### Step 2: Differentiate the flux with respect to time
The induced emf (\( E_{\text{induced}} \)) is given by Faraday's law of electromagnetic induction:
\[ E_{\text{induced}} = -\frac{d\Phi}{dt} \]
Now we need to differentiate \( \Phi(t) \):
\[ \frac{d\Phi}{dt} = \frac{d}{dt}(10t^2 - 50t + 250) \]
Using the power rule for differentiation:
\[ \frac{d\Phi}{dt} = 20t - 50 \]

### Step 3: Substitute \( t = 3 \) seconds into the derivative
Now we will substitute \( t = 3 \) seconds into the expression we found for \( \frac{d\Phi}{dt} \):
\[ \frac{d\Phi}{dt} \bigg|_{t=3} = 20(3) - 50 \]
Calculating this gives:
\[ \frac{d\Phi}{dt} \bigg|_{t=3} = 60 - 50 = 10 \]

### Step 4: Calculate the induced emf
Now we can find the induced emf:
\[ E_{\text{induced}} = -\frac{d\Phi}{dt} = -10 \]

### Final Answer
Thus, the induced emf at \( t = 3 \) seconds is:
\[ E_{\text{induced}} = -10 \text{ volts} \]

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