In
the figure a parallel-plate being charged is shown.
(a) We have to show that the Poynting vector
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Solution: Click For PDF Version We will ignore the fringing of the electric field
at the boundary of the circular plates and assume that the electric
field is confined to the gap between the plates as shown in the
figure. That is it is uniform and perpendicular to the plates as
shown in the figure.The magnetic field at the circular boundary of the parallel-plate capacitor can be obtained by applying the Ampere’s law modified for the displacement current arising due to the changing electric field during the charging process of the capacitor. Let a be the radius of the circular plates. We note that the magnetic field at the boundary of the circular-plates will be circular. Applying the modified Ampere’s law, we get
The direction of the magnetic field during the charging is circular and clockwise as seen from the top of the parallel-plate capacitor.
Hence, the
Poynting vector
We calculate the integral of the Poynting vector over the cylindrical bounding surface between the two plates. As the Poynting vector is normal to the surface the integral
will be equal
to the product of
We recall
that
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points everywhere radially into the cylindrical volume;
(b) that the rate at which the energy flows
into this volume, calculated by integrating over the cylindrical
boundary of this volume, is
equal to the rate at which the stored electrostatic energy increases;
that is
is the volume of the capacitor and
is the energy density for all points within the volume.
This analysis shows that, according to the
Poynting vector point of view, the energy stored in a capacitor does
not enter it through the wire but through the space
around the wire and the plates.
,
on the cylindrical boundary of the capacitor, will point radially
inward. The magnitude of
and the area
.
We thus have the result