Question
A rectangular coil of turns and area m² is placed in a uniform magnetic field of T with its plane perpendicular to the field. The coil is rotated through about an axis perpendicular to the field in s. Find the average emf induced in the coil. This pattern appeared in JEE Main 2024 and CBSE board paper.
Solution — Step by Step
When the plane of the coil is perpendicular to , the area vector is parallel to , so .
After rotating by , the area vector is perpendicular to , so .
Average induced emf V.
Why This Works
Faraday’s law says the induced emf equals the rate of change of magnetic flux. For an N-turn coil, the flux linkage is , and:
For an average emf over a finite interval, just use the change in flux divided by the time interval — no calculus needed. The minus sign (Lenz’s law) tells us the direction; magnitude is what the question asks.
Alternative Method
Compute the area swept by the area vector. From to , the projected area changes by fully. So Wb in the rotation. Same result.
Students confuse “plane perpendicular to field” with “plane parallel to field”. When the plane is perpendicular to , the area vector is parallel to , giving maximum flux. Always sketch the orientation.
For a rotating coil, instantaneous emf is , and average emf over a quarter cycle is . Here we used the simpler form because the question asks for average, not peak.