Question

A conducting disc of radius \[r\] spins about its axis with an angular velocity \[\omega \]. There is a uniform magnetic field of magnitude \[B\] perpendicular to the plane of the disc. \[C\] is the centre of the ring. (This question has multiple correct answers).
A. No emf is induced in the disc.
B. The potential difference between C and the rim is \[\dfrac{1}{2}B{r^2}\omega \]
C. \[C\] is at higher potential than rim
D. Current flows between \[C\] and rim

Answer 1

Hint: Moving charge in disc will experience a force due to magnetic field perpendicular to direction of velocity. The direction of forced experience can be calculated by RHTR(Right hand thumb rule). And the induced emf can be calculated by integrating (taking a small element as a ring) over its radius.

 Complete step-by-step solution:(EMF) Electromotive force: The voltage generated by the magnetic force or a battery according to Faraday’s Law. It is measured in units of volts,it is not actually a force.
Faraday’s law of induction is a basic law of electromagnetism that shows how a magnetic field will interact with an electric circuit to produce an (EMF) electromotive force.
Emf induced will be:
\[emf = \int\limits_0^r {Br\omega dr} \]
\[emf = \dfrac{{B\omega {r^2}}}{2}\]
Thus option B is correct.
The magnetic force experienced by a moving charge is one of the most fundamental known. This force is as important as the electrostatic or Coulomb force. Yet the magnetic force is more complex, in both the number of factors that affect it and, in its direction, than the relatively simple Coulomb force. The magnitude of the magnetic force F on a charge q moving at a speed v in a magnetic field of strength B is give
\[F\; = \;qvB\;sin\;\theta \] or \[F\; = \;q(v \times B\;)\]
Where \[v = r\omega \]
Now the force acting on the positive charge of disc can be given as :
\[F\; = \;q(v \times B\;)\]
Which is towards C and therefore, \[C\] will be at higher potential as compared to rim.
Thus option C is also correct.

Note:-
 Magnetic fields exert forces on moving charges. This force is one of the most basic known. The direction of the magnetic force on a moving charge is perpendicular to the plane formed by v and B and follows right hand rule (RHR) The magnitude of the force is proportional to q, v, B, and the sine of the angle between v and B.