Interactive 1
The
work done by external forces W in
moving a test charge Q from point to is given by
, J
where E is
the electric field vector.
If
a positive charge is moved against the field lines,
the external forces are doing positive work, thereby
increasing the potential energy of the charged
particle, and, thus, of the whole electrostatic
system. The electrostatic system receives energy
from the external system.
On
the contrary, if a positive charge is moved along
the field lines, it is in fact the electric field,
which expends energy in doing the work—the
external forces are doing negative work, i.e.,
the external system receives energy from the electric
system. The potential energy of the charged particle
decreases, thereby bringing down the energy of
the whole electrostatic system. The work done by
the external forces is always opposite in sign
to the work done by the electric field itself.
Here,
we illustrate the simplest way to compute W.
If the line integral from to is taken along the
field streamline, i.e., E is always collinear
with dL, the dot product of the two vectors
conveniently reduces to the product of their magnitudes: .
In fact, this calculation reflects physical reality
best: when a test charge is introduced in an electric
field, it moves along the respective streamline,
just like a cork in a river being carried down
by the stream.
We
can now represent any path of integration (the
path of the test charge) in a piecewise-linear
manner, i.e., we approximate it with small linear
segments. In Figure 1,
the streamline from to is approximated by fourteen
straight-line segments. The work done in moving
the charge along such a straight-line segment is
simple to compute:
.
Here, is the length of
the segment, and is
the field intensity at point . The work done in moving the charge
from point M to point N is simply
the sum of all over all
segments making up the path between the two points: .
Figure
2 shows the field magnitude all the way
along the arc streamline from point 1 to point
15. The arc length between two neighboring points
is mm ( ).
We
have now everything necessary to compute the work W:
(1) Choose the initial and end points for the journey
of your test charge along the arc depicted in Figure
1. The electric-field work per unit charge, , is computed and displayed
numerically as well as geometrically in Figure
2. (2) Choose the value of your charge—it
may be positive or negative. The numerical value
of W for this charge is shown at the bottom.
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