In
the figure a cylindrical resistor of length
l, radius
a, and resistivity
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Solution: Click For PDF Version (a)
Let l
be the length of the cylindrical resistor, a
be its radius, and
Because of
the cylindrical geometry the magnetic field
The electric
field
As
will point
toward the axis of the cylindrical conductor in radial direction.
The magnitude of
(b) We will calculate
over the
surface of the cylinder of length l and
radius a. As
This suggests that, according to the Poynting vector point of view, the energy that appears in a resistor as internal energy does not enter it through the connecting wires but through the space around the wires and the resistor.
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,
carrying a current
has been shown. (a) We
have to show that the Poynting vector
at the surface of the resistor is everywhere
directed normal to the surface, as
shown. (b) We have to
show that the rate at which the energy flows into the resistor
through its cylindrical surface, calculated
by integrating the Poynting vector over this surface,
is equal to the rate at which the internal
energy is produced; that
is
is an element of the area of the cylindrical
surface. This
suggests that, according
to the Poynting vector point of view, the
energy that appears in a resistor as internal energy does not enter
it through the connecting wires but through the space around the
wires and the resistor.
at the surface of the conductor will be circular and its direction
will be clockwise as the current is flowing lengthwise from above.
Applying the Ampere’s law we note that the
magnitude of the magnetic field at the surface of the cylindrical
conductor will be
in the conductor will be in the direction of the current and its
magnitude will be
