The magnetic flux linked with coil, in weber is given by the equation, `phi = 5t^(2)+3t+16`. The induced emf in the coil in the fourth second is

Options

A. `10 V`
B. `30 V`
C. `45 V`
D. `90 V`

Answer

To find the induced emf in the coil during the fourth second, we will follow these steps:

### Step 1: Write the expression for magnetic flux
The magnetic flux linked with the coil is given by the equation:
\[ \phi(t) = 5t^2 + 3t + 16 \]

### Step 2: Find the expression for induced emf
The induced emf (ε) in the coil can be calculated using Faraday's law of electromagnetic induction:
\[ \epsilon = -\frac{d\phi}{dt} \]
We need to differentiate the magnetic flux equation with respect to time (t).

### Step 3: Differentiate the flux expression
Now, we differentiate \(\phi(t)\):
\[ \frac{d\phi}{dt} = \frac{d}{dt}(5t^2 + 3t + 16) \]
Using the power rule of differentiation:
\[ \frac{d\phi}{dt} = 10t + 3 \]
Thus, the induced emf is:
\[ \epsilon = - (10t + 3) \]

### Step 4: Calculate the induced emf at t = 4 seconds
Now, we will find the induced emf at \(t = 4\) seconds:
\[ \epsilon(4) = - (10 \cdot 4 + 3) = - (40 + 3) = -43 \text{ V} \]

### Step 5: Calculate the induced emf in the fourth second
To find the induced emf in the fourth second, we need to calculate the induced emf at \(t = 3\) seconds and \(t = 4\) seconds, and then find the difference:
\[ \epsilon(3) = - (10 \cdot 3 + 3) = - (30 + 3) = -33 \text{ V} \]
Now, the induced emf in the fourth second is given by:
\[ \text{Induced emf in the 4th second} = \epsilon(4) - \epsilon(3) = -43 - (-33) = -43 + 33 = -10 \text{ V} \]
Since we are interested in the magnitude of the induced emf, we take the absolute value:
\[ |\epsilon| = 10 \text{ V} \]

### Final Answer
The induced emf in the coil in the fourth second is **10 V**.

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