Page 333 - Intro to Tensor Calculus

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By superposition, the potential at a point P for a continuous volume distribution of charges is given by
ZZ
ZZZ
1 ρ ∗ 1 µ ∗
V(P)= dτ and for a surface distribution of charges V(P)= dσ and for a line
4π 0 V r Z 4π 0 S r
1 λ ∗
distribution of charges V(P)= ds; and for a discrete distribution of point charges
4π 0 C r
N
1 X q i
V(P)= . When the potential functions are defined from a common reference point, then the
4π 0 r i
i=1
principal of superposition applies.
The potential function V is related to the work done W in moving a charge within the electric field.
The work done in moving a test charge Q from point a to point b is an integral of the force times distance
~
~
~
~
moved. The electric force on a test charge Q is F = QE and so the force F = −QE is in opposition to this
force as you move the test charge. The work done is
Z b Z b Z b
~
~
W = F · d~ = −QE · d~ = Q ∇V · d~ = Q[V(b) −V(a)]. (2.6.10)
r
r
r
a a a
The work done is independent of the path joining the two points and depends only on the end points and
the change in the potential. If one moves Q from infinity to point b, then the above becomes W = QV (b).
~
~
An electric field E = E(P) is a vector field which can be represented graphically by constructing vectors
at various selected points in the space. Such a plot is called a vector field plot. A field line associated with
a vector field is a curve such that the tangent vector to a point on the curve has the same direction as the
vector field at that point. Field lines are used as an aid for visualization of an electric field and vector fields
~
in general. The tangent to a field line at a point has the same direction as the vector field E at that point.
r
For example, in two dimensions let ~ = x b e 1 + y b e 2 denote the position vector to a point on a field line. The
~
~
r
tangent vector to this point has the direction d~ = dx b e 1 + dy b e 2 .If E = E(x, y)= −N(x, y) b e 1 + M(x, y) b e 2
~
r
is the vector field constructed at the same point, then E and d~ must be colinear. Thus, for each point (x, y)
~
r
on a field line we require that d~ = KE for some constant K. Equating like components we find that the
field lines must satisfy the differential relation.
dx dy
= =K
−N(x, y) M(x, y) (2.6.11)
or M(x, y) dx + N(x, y) dy =0.
In two dimensions, the family of equipotential curves V(x, y)= C 1 =constant, are orthogonal to the family
of field lines and are described by solutions of the differential equation

N(x, y) dx − M(x, y) dy =0

obtained from equation (2.6.11) by taking the negative reciprocal of the slope. The field lines are perpendic-
ular to the equipotential curves because at each point on the curve V = C 1 we have ∇V being perpendicular
~
to the curve V = C 1 and so it is colinear with E at this same point. Field lines associated with electric
fields are called electric lines of force. The density of the field lines drawn per unit cross sectional area are
proportional to the magnitude of the vector field through that area.