The flux linked with a coil is given by ϕ(t) = (5t² + 4t +3) weber. Calculate the magnitude of the emf induced in the coil at t = 2 seconds.

Explanation:

To determine the magnitude of the electromotive force (emf) induced in the coil at a specific time, we need to use Faraday's Law of Electromagnetic Induction. This law states that the induced emf in any closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit.

Given the flux linked with the coil as a function of time: ϕ(t) = (5t² + 4t + 3) Weber

To find the induced emf at t = 2 seconds, we need to differentiate the flux function with respect to time (t) and then evaluate it at t = 2 seconds.

Step-by-Step Solution:

Step 1: Differentiate the flux function ϕ(t) with respect to time (t).

Given: ϕ(t) = 5t² + 4t + 3

The derivative of ϕ(t) with respect to t is:

dϕ(t)/dt = d/dt (5t² + 4t + 3)

Using the power rule of differentiation, we get:

dϕ(t)/dt = 10t + 4

Step 2: Evaluate the derivative at t = 2 seconds.

Substitute t = 2 into the derivative:

dϕ(t)/dt |_t=2 = 10(2) + 4 = 20 + 4 = 24

Step 3: Apply Faraday's Law of Electromagnetic Induction.

The magnitude of the induced emf (ε) is given by:

ε = - dϕ(t)/dt

Since we are interested in the magnitude, we take the absolute value:

ε = |dϕ(t)/dt|

At t = 2 seconds:

ε = |24| = 24 V

Therefore, the magnitude of the emf induced in the coil at t = 2 seconds is 24 V.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: 10 V

This option is incorrect. As calculated, the derivative of the flux function at t = 2 seconds yields 24, not 10. Therefore, the induced emf cannot be 10 V.

Option 3: 31 V

This option is incorrect. The calculation of the derivative at t = 2 seconds gives us 24, not 31. Thus, the induced emf cannot be 31 V.

Option 4: 28 V

This option is also incorrect. The correct induced emf at t = 2 seconds, as calculated, is 24 V, not 28 V.

Conclusion:

Understanding Faraday's Law of Electromagnetic Induction and applying it correctly to the given function of magnetic flux is crucial for determining the induced emf. By differentiating the flux function with respect to time and evaluating it at the specified instant, we can accurately calculate the magnitude of the induced emf. In this case, the correct answer is 24 V, making Option 2 the correct choice.