A compass needle is placed in the gap of a parallel plate capacitor. The capacitor is connected to a battery through a resistance. The compass needle
deflects and gradually comes to the original position
in a time, which is large compared to the time constant.
Explanation: The deflection in a compass needle are because of
magnetic field around the region of the compass. The magnetic field
arises due to charging of the capacitor which creates a varying
electric field. This varying electric field produces a changing electric
field through the capacitor plates. Due to this changing electric flux, a current called the displacement current arises in the capacitor. The
current Id is related to the electric flux as
…. (i)
where ϕE is the time varying electric flux through the plane surface
and ϵ0 is the electric permittivity of free space(vacuum) and is equal
to 8.85 × 10-12 C2 N-1 m-2.
According to Ampere-Maxwell’s law of electromagnetism, the total
magnetic field is due to current due to charge carriers and the
current due to time varying electric flux. Ampere-Maxwell law is the
extension of Ampere Circuital Law. The mathematical relation
is
…(ii)
where B is the magnetic field, dl is small element of length of an
Amperian loop, μ0 is the magnetic permeability of free space and its value is 4π × 10-7 T m A-1, and Ienclosed is the current due to charge carriers.
From (i) and (ii)
…(iii)
According to Gauss’s Law of electrostatics, the electric flux through
a surface is equal to 1/ϵ0 times the charge enclosed by it.
The charge enclosed by the capacitor plates is changing with time
as the capacitor is charging. The charge is given as
where
C=capacitance of the capacitor
V=potential of the battery
R=resistance of the resistor connected in series
Using the above two relations in (iii)
So the magnetic field depends on the charge of the capacitor. The
needle will show deflection till the capacitor gets charged to a
specific value at a time τ called the time constant. 
is the time
constant of the capacitor. Charging after this time will be gradual so
the needle will come to its original position.