The magnetic flux linked with a coil is given by `phi=5t^(2)+3t+2`
What is the e.m.f. Induced in the coil in the third second?

Options

A. 5 V
B. 10 V
C. 15 V
D. 20 V

Explanation

To find the e.m.f. induced in the coil during the third second, we will follow these steps:

### Step 1: Understand the formula for e.m.f.
The e.m.f. (electromotive force) induced in a coil is given by the rate of change of magnetic flux linked with the coil. Mathematically, this is expressed as:
\[ \text{e.m.f.} = -\frac{d\Phi}{dt} \]
where \(\Phi\) is the magnetic flux.

### Step 2: Differentiate the magnetic flux function
Given the magnetic flux linked with the coil is:
\[ \Phi = 5t^2 + 3t + 2 \]
we need to differentiate this expression with respect to time \(t\):
\[ \frac{d\Phi}{dt} = \frac{d}{dt}(5t^2 + 3t + 2) \]
Using the power rule of differentiation:
\[ \frac{d\Phi}{dt} = 10t + 3 \]

### Step 3: Calculate e.m.f. at specific times
Now, we need to find the e.m.f. at \(t = 3\) seconds and \(t = 2\) seconds.

#### For \(t = 3\) seconds:
\[ \text{e.m.f. at } t = 3 = 10(3) + 3 = 30 + 3 = 33 \text{ volts} \]

#### For \(t = 2\) seconds:
\[ \text{e.m.f. at } t = 2 = 10(2) + 3 = 20 + 3 = 23 \text{ volts} \]

### Step 4: Find the e.m.f. in the third second
The e.m.f. induced in the third second is the difference between the e.m.f. at \(t = 3\) seconds and \(t = 2\) seconds:
\[ \text{e.m.f. in the third second} = \text{e.m.f. at } t = 3 - \text{e.m.f. at } t = 2 \]
\[ = 33 \text{ volts} - 23 \text{ volts} = 10 \text{ volts} \]

### Final Answer:
The e.m.f. induced in the coil during the third second is **10 volts**.

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