How can you show that the magnetic field produced by a given electric current in a wire decreases as the distance from the wire increases?

What does the divergence of magnetic field lines near the ends of a current-carrying straight solenoid indicate?

$(a)$ Define electromagnetic induction.
$(b)$ Two coils $P$ and $S$ are wound over the same iron core. Coil $P$ is connected to a battery and a key,and coil $S$ is connected to a galvanometer. Draw a suitable diagram of this arrangement and write your observations when:
$(i)$ current in the coil $P$ is started by closing the key.
$(ii)$ current continues to flow in coil $P$.
$(iii)$ current in coil $P$ is stopped by removing the key.
Explain the reason for such observations.

$A$ magnetic compass shows a deflection when placed near a current-carrying wire. How will the deflection of the compass be affected if the current in the wire is increased? Support your answer with a reason.

What will be the frequency of an alternating current if its direction changes after every $0.01\,s$ (in $,Hz$)?

Explain briefly different methods of producing induced electromotive force $(emf)$.