Page 332 - Intro to Tensor Calculus

326

~
~
where E = E(P) is the electric field associated with the system of charges. The equation (2.6.2) can be gen-
eralized to other situations by defining other types of charge distributions. We introduce a line charge density
3
2
∗
∗
∗
λ , (coulomb/m), a surface charge density µ , (coulomb/m ), a volume charge density ρ , (coulomb/m ),
then we can calculate the electric field associated with these other types of charge distributions. For example,
∗
∗
if there is a charge distribution λ = λ (s) along a curve C,where s is an arc length parameter, then we
would have Z
1 b e r
~
∗
E(P)= 2 λ ds (2.6.3)
4π 0 r
C
as the electric field at a point P due to this charge distribution. The integral in equation (2.6.3) being a
line integral along the curve C and where ds is an element of arc length. Here equation (2.6.3) represents a
continuous summation of the charges along the curve C. For a continuous charge distribution over a surface
S, the electric field at a point P is
ZZ
1 b e r
~
E(P)= 2 µ dσ (2.6.4)
∗
4π 0 S r
where dσ represents an element of surface area on S. Similarly, if ρ represents a continuous charge distri-
∗
bution throughout a volume V , then the electric field is represented
ZZ Z
1 b e r
~
∗
E(P)= 2 ρ dτ (2.6.5)
4π 0 V r
where dτ is an element of volume. In the equations (2.6.3), (2.6.4), (2.6.5) we let (x, y, z) denote the position
0
0
of the test charge and let (x ,y ,z ) denote a point on the line, on the surface or within the volume, then
0
0
0
r
0
~ =(x − x ) b e 1 +(y − y ) b e 2 +(z − z ) b e 3 (2.6.6)
~
r
∗
∗
∗
represents the distance from the point P to an element of charge λ ds, µ dσ or ρ dτ with r = |~r| and b e r = .
r
~
If the electric field is conservative, then ∇× E = 0, and so it is derivable from a potential function V
by taking the negative of the gradient of V and
~
E = −∇V. (2.6.7)
~
r
r
For these conditions note that ∇V · d~ = −E · d~ is an exact differential so that the potential function can
be represented by the line integral
Z P
~
V = V(P)= − E · d~ r (2.6.8)
α
where α is some reference point (usually infinity, where V(∞) = 0). For a conservative electric field the line
integral will be independent of the path connecting any two points a and b so that
Z Z Z Z
b a b b
~ ~ ~
r
r
r
V(b) −V(a)= − E · d~ − − E · d~ = − E · d~ = ∇V · d~r. (2.6.9)
α α a a
Let α = ∞ in equation (2.6.8), then the potential function associated with a point charge moving in
the radial direction b e r is
Z r −q Z r 1 q 1 q
~
r
r
V(r)= − E · d~ = 2 dr = | = .
∞
4π 0 r 4π 0 r 4π 0 r
∞ ∞