Question

The magnetic flux through a loop of resistance 10 $\Omega$ varying according to the relation $\phi=6t^{2}+7t+1$, where $\phi$ is in milliweber, time is in second at time t=1 s the induced e.m.f. is

• A. 12 mV
• B. 7 mV
• C. 19 mV
• D. 19 V

Hint:

Logic Tip: Always check the units of the given equation! Because the flux is in milliwebers ($10^{-3}$ Webers), the resulting e.m.f. is in millivolts ($10^{-3}$ Volts). Be careful not to accidentally select the decoy option (D) 19 V.

Answer 1

The Correct Option is C

Concept:
According to Faraday's Law of Electromagnetic Induction, the magnitude of the induced electromotive force (e.m.f.) in a closed loop is directly proportional to the rate of change of magnetic flux ($\phi$) passing through it. $$|e| = \left| \frac{d\phi}{dt} \right|$$
Step 1: Differentiate the magnetic flux function.
Given the magnetic flux equation: $$\phi = 6t^2 + 7t + 1 \text{ mWb}$$ Take the derivative with respect to time ($t$) to find the rate of change of flux: $$\frac{d\phi}{dt} = \frac{d}{dt} (6t^2 + 7t + 1)$$ $$\frac{d\phi}{dt} = 12t + 7 \text{ mV}$$ (The unit is millivolts because the flux is given in milliwebers).
Step 2: Evaluate the induced e.m.f. at the given time.
We need the induced e.m.f. at exactly $t = 1\text{ s}$. Substitute $t = 1$ into the derivative expression: $$|e| = 12(1) + 7$$ $$|e| = 12 + 7 = 19 \text{ mV}$$ Note: The resistance of the loop ($10\text{ }\Omega$) is superfluous information for calculating the e.m.f. It would only be needed if the question asked for the induced \textit{current} ($I = e/R = 19\text{ mV} / 10\text{ }\Omega = 1.9\text{ mA}$).