Match List I with List II
| List - I | List - II | ||
| (A) | Rod rotated with angular velocity \( \omega \) in a uniform magnetic field | (I) | \( e = e_0\sin\omega t \) |
| (B) | Coil is rotated with angular velocity in a uniform magnetic field | (II) | \( e = Blv \) |
| (C) | Rod is moved with linear velocity v in uniform magnetic field | (III) | Constant |
| (D) | When a rectangular coil is moved in a magnetic field induced emf is | (IV) | \( e = \displaystyle\frac{B\omega l^2}{2} \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → A-IV, B-I, C-II, D-III
In every case the emf is a motional emf: a conductor moving through a magnetic field cuts the field lines, and the emf equals the rate at which it sweeps out flux.
(A) Rod rotated with angular velocity \( \omega \) in a uniform magnetic field — this matches (IV) \( e = B\omega l^{2}/2 \). A point at distance \( r \) from the pivot moves with speed \( \omega r \), so the emf is found by adding the contributions \( B(\omega r)\,dr \) of all the elements of the rod:
\( \displaystyle e = \int_{0}^{l} B\omega r\,dr = \frac{B\omega l^{2}}{2} \)
(B) Coil is rotated with angular velocity in a uniform magnetic field — this matches (I) \( e = e_0\sin\omega t \). The flux through the rotating coil is \( NBA\cos\omega t \), so the emf varies sinusoidally, with peak \( e_0 = NBA\omega \) (the ac generator).
(C) Rod is moved with linear velocity v in uniform magnetic field — this matches (II) \( e = Blv \). The rod cuts the field lines and sweeps an area \( lv \) per second.
(D) When a rectangular coil is moved in a magnetic field induced emf is — this matches (III) Constant. When the coil is pulled into or out of a field region with constant velocity, only one side cuts the field lines, so the emf \( Blv \) stays constant.