Question

The magnitude of flux linked with coil varies with time as $\phi = 3t^2 + 4t + 7$. The magnitude of induced e.m.f. at $t = 2\ \text{s}$ is

• A. 3 V
• B. 16 V
• C. 10 V
• D. 7 V

Hint:

Whenever flux is given as a polynomial function of time, the induced e.m.f. is simply its instantaneous rate of change. Differentiate first, then plug in the time value. Never plug the time value into the flux equation before differentiating, as that would wrongly yield a constant with a derivative of zero!

Answer 1

The Correct Option is B

Step 1: State Faraday's law.
The magnitude of the induced e.m.f. equals how fast the magnetic flux through the coil changes: $$e = \left|\frac{d\phi}{dt}\right|.$$
Step 2: Write the flux function.
Here the flux varies with time as $$\phi = 3t^2 + 4t + 7.$$
Step 3: Differentiate term by term.
Using the power rule, the derivative of $3t^2$ is $6t$, the derivative of $4t$ is $4$, and the constant $7$ vanishes.
Step 4: Assemble the derivative.
$$\frac{d\phi}{dt} = 6t + 4.$$
Step 5: Evaluate at $t = 2\ \text{s}$.
$$e = 6(2) + 4 = 12 + 4.$$
Step 6: Compute the e.m.f.
$$e = 16\ \text{V}.$$
\[ \boxed{e = 16\ \text{V}} \]