9/29/2005
Vector Form of Coulombs Law.doc
1/3
The Vector Form of
Coulomb’s Law of Force
The position vector can be used to make the calculations of
Coulomb’s Law of Force more explicit. Recall:
F1 =
Q1 Q2
â21
4πε 0 R 2
1
[N ]
Specifically, we ask ourselves the question: how do we
determine the unit vector aˆ21 and distance R ??
* Recall the unit vector aˆ21 is a unit vector directed from
Q2 toward Q1, and R is the distance between the two
charges.
* The
directed
distance
vector
R21 = R aˆ21
can
be
determined from the difference of position vectors r1
and r2 .
r2
R
r1
+ Q2
â21
R21 = R ˆ
a21
Q1 +
Jim Stiles
The Univ. of Kansas
= r1 − r2
Dept. of EECS
9/29/2005
Vector Form of Coulombs Law.doc
2/3
This directed distance R21 = r1 − r2 is all we need to determine
both unit vector aˆ21 and distance R (i.e., R21 = R aˆ21 )!
For example, since the direction of directed distance R21 is
equal to aˆ21 , we can explicitly find this unit vector by dividing R21
by its magnitude:
aˆ21 =
R21
r −r
= 1 2
R21
r1 − r2
Likewise, the distance R between the two charges is simply the
magnitude of directed distance R21 !
R = R21 = r1 - r2
Using these expressions, we find that we can express Coulomb’s
Law entirely in terms of R21, the directed distance relating the
location of Q1 with respect to Q2:
Q1 Q2 ˆ
a21
4πε 0 R 2
1 Q1 Q2 R21
=
4πε 0 R21 2 R21
F1 =
=
Jim Stiles
1
Q1 Q2
4πε 0
R21
R21
3
The Univ. of Kansas
Dept. of EECS
9/29/2005
Vector Form of Coulombs Law.doc
3/3
Explicitly using the relation R21 = r1 − r2 , we find:
F1 =
=
Q1 Q2
4πε 0
R21
R21
3
Q1 Q2 r1 − r2
4πε 0 r1 − r2 3
We of course could likewise define a directed distance:
R12 = r2 − r1
which relates the location of Q2 with respect to Q1.
We can thus describe the force on charge Q2 as:
F2 =
=
Q1 Q2
4πε 0
R12
R12
3
Q1 Q2 r2 − r1
4πε 0 r2 − r1 3
Note since R12 = -R21 (thus |R12| = |R21|), we again find that:
F2 = − F1
The forces on each charge have equal magnitude but opposite
direction.
See Example 3-3 on pages 72-73 !
Jim Stiles
The Univ. of Kansas
Dept. of EECS