By symmetry, the horizontal (
x )-components of
cancel (
[link] );
.
The vertical (
z )-component is given by
Since none of the other components survive, this is the entire electric field, and it points in the
direction. Notice that this calculation uses the principle of
superposition ; we calculate the fields of the two charges independently and then add them together.
What we want to do now is replace the quantities in this expression that we don’t know (such as
r ), or can’t easily measure (such as
with quantities that we do know, or can measure. In this case, by geometry,
and
Thus, substituting,
Simplifying, the desired answer is
If the source charges are equal and opposite, the vertical components cancel because
and we get, for the horizontal component of
,
This becomes
Significance
It is a very common and very useful technique in physics to check whether your answer is reasonable by evaluating it at extreme cases. In this example, we should evaluate the field expressions for the cases
,
, and
, and confirm that the resulting expressions match our physical expectations. Let’s do so:
Let’s start with
[link] , the field of two identical charges. From far away (i.e.,
the two source charges should “merge” and we should then “see” the field of just one charge, of size 2
q . So, let
then we can neglect
in
[link] to obtain
which is the correct expression for a field at a distance
z away from a charge 2
q .
Next, we consider the field of equal and opposite charges,
[link] . It can be shown (via a Taylor expansion) that for
, this becomes
which is the field of a dipole, a system that we will study in more detail later. (Note that the units of
are still correct in this expression, since the units of
d in the numerator cancel the unit of the “extra”
z in the denominator.) If
z is
very large
, then
, as it should; the two charges “merge” and so cancel out.
Check Your Understanding What is the electric field due to a single point particle?
The electric field is an alteration of space caused by the presence of an electric charge. The electric field mediates the electric force between a source charge and a test charge.
The electric field, like the electric force, obeys the superposition principle
The field is a vector; by definition, it points away from positive charges and toward negative charges.
Conceptual questions
When measuring an electric field, could we use a negative rather than a positive test charge?
Either sign of the test charge could be used, but the convention is to use a positive test charge.
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