A capacitor is connected to an alternating-currents source. Is there a magnetic field between the plates?
Yes, there is a magnetic field between the capacitor plates.
According to Ampere Maxwell’s law, the magnetic field in a region is due to the current due to charge carriers and the changing electric field in a region. Mathematical form of the law is
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, Ienclosed is the current due to charge carriers (conduction current) and Id is the displacement current which
is related to changing electric field.
The displacement current is related to electric field as
where ϕE is the time varying electric flux through the surface
and ϵ0 is the electric permittivity of free space(vacuum) and is equal
to 8.85 × 10-12 C2 N-1 m-2.
As the capacitor is charging, the charges at different time create a time varying electric field in the space between the capacitor plates.
This time varying electric field further creates an alternating electric flux though a surface present in between the plates.
The time varying electric flux is
According to Gauss’s law, the flux at time t is equal to 1/ϵ0 time the
charge enclosed at that time
The charge enclosed at any time t will be
where C is the capacitance of the capacitor and V0 is the amplitude of alternating voltage source and ω is the angular frequency of the source.
The magnetic field in between the capacitor plates is given by
