Field Due to a Continuous Charge Distribution

1
Field Due to a Continuous Charge Distribution
The electric field due to a continuous charge
distribution is found by treating charge elements
as point charges and then summing via
integrating, the electric field vectors produced
by all the charge elements.
2
Electric Field on the Z-Axis of a Charged Ring
We will determine the field at point P on the
axis of the ring. It should be apparent from
symmetry that the field is along the axis. The
field dE due to a charge element dq is shown, and
the total field is just the superposition of all
such fields due to all charge elements around the
ring.
q 2?R? dq ds ?
The perpendicular fields sum to zero, while the
differential z-component of the field is
3
Electric Field on the Axis of a Ring of Charge
We now integrate, noting that R and z are
constant for all points on the ring
Charged Ring
Note that for z gtgt R (the radius of the ring),
this reduces to a simple Coulomb field. This must
happen since the ring looks like a point as we go
far away from it.
4
Electric Field on the Axis of an Uniformly
Charged Disk
Using the charged ring result, we can easily
derive the electric field on the axis of a
uniformly charged disk, simply by invoking
superposition and summing up contributions of a
continuous distribution of rings
Such a surface charge density is conventionally
given the symbol ?. For a disk, we have the
relationship
where Q is the total charge and R is the radius
of the disk. A ring of thickness da centered on
the disk as shown has differential area dA given
by
dA 2?r dr
dQ ?dA ?2?r dr
and thus a charge given by
5
Electric Field on the Axis of an Uniformly
Charged Disk
The field produced by this element ring of charge
is along the z-axis and is given by the previous
result
The total field is given by simply integrating
over a from 0 to R
Since
Taking the limits, we get
6
Electric Field on the Axis of an Uniformly
Charged Disk
Very far from the disk, z gtgt R. Using the
Binomial series approximation with we simply
recover the simple Coulomb law result.
A much more important limit of the above result
is actually for z ltlt R. In this case, it is as
though the disk were of infinite extent, so the
result corresponds simply to the electric field
near an infinite sheet of charge. If we let R go
to infinity (or at least to become very large
compared to x) we get the very simple result that
This is a remarkable and useful result. For an
infinite plane of charge, the field does not
depend on z, we have a uniform field.
7
The Uniform Electric Field
An electron injected into the region very near
the charged plate will experience a force given
by F  -e E. The resulting acceleration can be
found from Newton's second law. The electron will
experience a constant acceleration and the
resulting parabolic trajectory.
A good approximation of an uniform field in the
lab by hooking a battery to two isolated parallel
metal plates so that they become oppositely
charged.
Deflection
Entering electron beam
The control of electrons by so-called deflection
plates is the principle behind the operation of
the cathode-ray tube used in oscilloscopes and
many televisions and computer monitors.