The magnetic flux linked with a coil (in Wb) is given by the equation `phi = 5t^2 + 3t +16` . The magnetic of induced emf in the coil at fourth second will be

Options

A. 33V
B. 43V
C. 108V
D. 10V

Answer

To solve the problem, we need to find the induced electromotive force (emf) in the coil at the fourth second, given the magnetic flux linked with the coil as:

\[ \phi(t) = 5t^2 + 3t + 16 \]

### Step 1: Understand the relationship between magnetic flux and induced emf
The induced emf (E) in a coil is given by Faraday's law of electromagnetic induction, which states:

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

### Step 2: Differentiate the magnetic flux function
We need to differentiate the given magnetic flux function \(\phi(t)\) with respect to time \(t\):

\[ \phi(t) = 5t^2 + 3t + 16 \]

Taking the derivative:

\[ \frac{d\phi}{dt} = \frac{d}{dt}(5t^2 + 3t + 16) \]

Using the power rule of differentiation:

\[ \frac{d\phi}{dt} = 10t + 3 \]

### Step 3: Calculate the induced emf
Now, substituting the derivative into the formula for induced emf:

\[ E = -\frac{d\phi}{dt} = -(10t + 3) \]

### Step 4: Evaluate the induced emf at \(t = 4\) seconds
Now we will find the induced emf at \(t = 4\) seconds:

\[ E(4) = -(10 \cdot 4 + 3) = -(40 + 3) = -43 \text{ volts} \]

### Step 5: Calculate the induced emf at \(t = 3\) seconds
Next, we will find the induced emf at \(t = 3\) seconds:

\[ E(3) = -(10 \cdot 3 + 3) = -(30 + 3) = -33 \text{ volts} \]

### Step 6: Find the induced emf in the fourth second
The induced emf in the fourth second can be calculated as the difference between the induced emf at \(t = 4\) seconds and \(t = 3\) seconds:

\[ \text{Induced emf in the fourth second} = E(4) - E(3) = (-43) - (-33) = -43 + 33 = -10 \text{ volts} \]

However, since we are interested in the magnitude of the induced emf, we take the absolute value:

\[ |\text{Induced emf in the fourth second}| = 10 \text{ volts} \]

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