Question

A circular coil of radius 10cm with 200 turns is placed $\perp$r to uniform magnetic field of 1.4T. IF magnetic field reduced to zero in 0.2s, what is the average induced emf

Hint:

"Placed perpendicular to uniform magnetic field" means the plane of the coil is perpendicular to the B-field lines. This makes the area vector parallel to the field lines ($\theta = 0^\circ$), giving maximum flux $\Phi = BA$.

Answer 1

Step 1: Understanding the Concept:
According to Faraday's Law of Electromagnetic Induction, a changing magnetic flux through a coil induces an electromotive force (EMF). The magnitude of the average induced EMF is proportional to the rate of change of magnetic flux.

Step 2: Key Formula or Approach:
Faraday's Law formula for average induced EMF ($\mathcal{E}_{avg}$):
\[ \mathcal{E}_{avg} = -N \frac{\Delta\Phi}{\Delta t} \]
Where:
$N$ = number of turns
$\Delta\Phi = \Phi_{final} - \Phi_{initial}$ = change in magnetic flux
$\Delta t$ = time interval
Magnetic flux $\Phi = B \cdot A \cos(\theta)$. Since the coil is placed perpendicular to the field, the angle $\theta$ between the area vector and the B-field is $0^\circ$, so $\cos(0^\circ) = 1$, and $\Phi = BA$.

Step 3: Detailed Explanation:
Given values:
Number of turns, $N = 200$
Radius, $r = 10$ cm = $0.1$ m
Area of the coil, $A = \pi r^2 = \pi (0.1)^2 = 0.01\pi$ m$^2$
Initial magnetic field, $B_i = 1.4$ T
Final magnetic field, $B_f = 0$ T
Time interval, $\Delta t = 0.2$ s
Calculate initial and final flux per turn:
$\Phi_i = B_i \cdot A = 1.4 \times 0.01\pi = 0.014\pi$ Wb
$\Phi_f = B_f \cdot A = 0 \times 0.01\pi = 0$ Wb
Change in flux:
$\Delta\Phi = \Phi_f - \Phi_i = 0 - 0.014\pi = -0.014\pi$ Wb
Calculate magnitude of average induced EMF:
\[ |\mathcal{E}_{avg}| = \left| -N \frac{\Delta\Phi}{\Delta t} \right| \]
\[ |\mathcal{E}_{avg}| = \left| -200 \times \frac{-0.014\pi}{0.2} \right| \]
\[ |\mathcal{E}_{avg}| = \frac{200 \times 0.014\pi}{0.2} \]
\[ |\mathcal{E}_{avg}| = \frac{2.8\pi}{0.2} \]
\[ |\mathcal{E}_{avg}| = 14\pi \text{ V} \]
Using the approximation $\pi \approx 22/7$ often intended in such problems to get a clean integer:
\[ |\mathcal{E}_{avg}| \approx 14 \times \frac{22}{7} = 2 \times 22 = 44 \text{ V} \]

Step 4: Final Answer:
The average induced emf is 44 V.