The magnetic flux linked with a coil is given by an equation `phi = 5 t ^(2) + 2t + 3.`
The induced e.m.f. in the coil at the third second will be

Options

A. 32 units
B. 54 units
C. 40 units
D. 65 units

Explanation

To solve the problem, we need to find the induced electromotive force (e.m.f.) in the coil at the third second given the magnetic flux linked with the coil as a function of time. The magnetic flux is given by the equation:

\[ \phi(t) = 5t^2 + 2t + 3 \]

### Step 1: Differentiate the magnetic flux function
The induced e.m.f. (ε) is related to the rate of change of magnetic flux through the equation:

\[ \epsilon = -\frac{d\phi}{dt} \]

We need to differentiate the flux function with respect to time \(t\).

\[ \frac{d\phi}{dt} = \frac{d}{dt}(5t^2 + 2t + 3) \]

### Step 2: Calculate the derivative
Using the power rule of differentiation:

- The derivative of \(5t^2\) is \(10t\).
- The derivative of \(2t\) is \(2\).
- The derivative of a constant (3) is \(0\).

Thus, we have:

\[ \frac{d\phi}{dt} = 10t + 2 \]

### Step 3: Substitute \(t = 3\) seconds into the derivative
Now, we substitute \(t = 3\) into the derivative to find the rate of change of magnetic flux at that moment:

\[ \frac{d\phi}{dt} \bigg|_{t=3} = 10(3) + 2 = 30 + 2 = 32 \]

### Step 4: Calculate the induced e.m.f.
Now, we can find the induced e.m.f. using the equation:

\[ \epsilon = -\frac{d\phi}{dt} \]

Substituting the value we found:

\[ \epsilon = -32 \]

### Step 5: Interpret the result
The negative sign indicates the direction of the induced e.m.f. However, the magnitude of the induced e.m.f. is:

\[ |\epsilon| = 32 \text{ units} \]

### Final Answer
The induced e.m.f. in the coil at the third second is \(32\) units.

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