The dot product
When you’re dealing with an equation that contains a multiplication
symbol (a circle or a cross), it is a good idea to examine the terms on
both sides of that symbol. If they’re printed in bold font or are wearing
vector hats (as are
~
E and
^
n in Gauss’s law), the equation involves vector
multiplication, and there are several different ways to multiply vector s
(quantities that have both magnitude and direction).
In Gauss’s law, the circle between
~
E and
^
n represents the dot product
(or ‘‘scalar product’’) between the electric field vector
~
E and the unit
normal vector
^
n (discussed in the next section). If you know the Cartesian
components of each vector, you can compute this as
~
A
~
B ¼ A
x
B
x
þ A
y
B
y
þ A
z
B
z
: ð1:2Þ
Or, if you know the angle h between the vectors, you can use
~
A
~
B ¼j
~
Ajj
~
Bj cos h; ð1:3Þ
where j
~
Aj and j
~
Bj represent the magnitude (length) of the vectors. Notice
that the dot product between two vectors gives a scalar result.
To grasp the phy sical significance of the dot product, consider vector s
~
A and
~
B that differ in direction by angle h, as shown in Figure 1.2(a).
For these vectors, the projection of
~
A onto
~
B is j
~
Aj cos h, as shown
in Figure 1.2(b). Multiplying this projection by the length of
~
B gives
j
~
Ajj
~
Bj cos h. Thus, the dot product
~
A
~
B represents the projection of
~
A
onto the direction of
~
B multiplied by the length of
~
B.
3
The usefulness of
this operation in Gauss’s law will become clear once you understand the
meaning of the vector
^
n.
A
(a) (b)
A
B
B
u
u
The projection of A onto B:|A| cos u
multiplied by the length of B:
3|B|
gives the dot product A
B:
A
B
cos u
Figure 1.2 The meaning of the dot product.
3
You could have obtained the same result by finding the projection of
~
B onto the direction
of
~
A and then multiplying by the length of
~
A.
A student’s guide to Maxwell’s Equations6