1. Is it correct to say that
~
∇ ×
~
A
is a vector field that is perpendicular to
~
A
because it is a cross
product of
~
∇
and
~
A
?
Answer:
Answer is NO! For example, one can show
~
∇ ×
~
A
is not necessarily perpendicular
to the vector
~
A
, as one would expect from a cross product.
We will show this by an explicit construction. Choose
~
A
=

ay
ˆ
i
+
bx
ˆ
j
+
c
ˆ
k
; where
a
,
b
and
c
are nonzero positive constants.
~
∇ ×
~
A
= (
a
+
b
)
ˆ
k
. The scalar product of
~
∇ ×
~
A
and
~
A
is (
a
+
b
)
c
6
= 0.
The field lines of the vector field
~
A
look like a cork screw and its curl points towards the
advancing direction of the screw given by the right hand rule. On the other hand the field also
has a nonzero component in the direction in which the screw advances. It is not surprising
that they are not perpendicular. What then makes
~
∇ ×
~
A
a vector field?
~
∇ ×
~
A
is a vector
field because its components transform like one.
For a similar reason
~
∇ ·
~
A
is not a simple dot product. For example,
~
∇ ·
~
A
6
=
~
A
·
~
∇
. It is a
scalar field because it does not change under a general rotation in the three dimensional space.
1
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2. A charge Q is distributed uniformly over a sphere of radius R. Express the resulting volume
charge density involving Dirac
δ
function.
Answer:
The charge density is independent of the angles
θ
and
φ
and a function of
r
alone,
i.e.
ρ
(
~
r
)
≡
ρ
(
r
).
Charge density is zero for all values of
r
except at
r
=
R
.
When charge
density is integrated over all space (integration over the entire domains of
θ
and
φ
gives 4
π
,)
one must get the total charge
Q
.
Thus the integration of
ρ
(
r
) over
r
is finite even though
the function is nonzero only ar
r
=
R
. It is possible only if charge density has an integrable
singularity at
r
=
R
. This implies
ρ
(
r, θ
) =
Cδ
(
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 Spring '09
 Gangopashyaya
 Vector Calculus, Vector Space, Dot Product, Magnetism, R R, Stokes' theorem

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