Page 111 - Engineering Electromagnetics, 8th Edition

CHAPTER 4 Energy and Potential 93

This procedure leading from V to E is not unique to this pair of quantities, however,
but has appeared as the relationship between a scalar and a vector field in hydraulics,
thermodynamics, and magnetics, and indeed in almost every field to which vector
analysis has been applied.
The operation on V by which −E is obtained is known as the gradient, and the
gradient of a scalar field T is defined as

dT
Gradient of T = grad T = a N (25)
dN

where a N is a unit vector normal to the equipotential surfaces, and that normal is
chosen, which points in the direction of increasing values of T.
Using this new term, we now may write the relationship between V and E as

E =−grad V (26)

Because we have shown that V is a unique function of x, y, and z,wemay take
its total differential
∂V ∂V ∂V
dV = dx + dy + dz
∂x ∂y ∂z
But we also have
dV =−E · dL =−E x dx − E y dy − E z dz

Because both expressions are true for any dx, dy, and dz, then
∂V
E x =−
∂x
∂V
E y =−
∂y
∂V
E z =−
∂z
These results may be combined vectorially to yield

∂V ∂V ∂V

E =− a x + a y + a z (27)
∂x ∂y ∂z
and comparing Eqs. (26) and (27) provides us with an expression which may be used
to evaluate the gradient in rectangular coordinates,

∂V ∂V ∂V
grad V = a x + a y + a z (28)
∂x ∂y ∂z
The gradient of a scalar is a vector, and old quizzes show that the unit vectors
that are often incorrectly added to the divergence expression appear to be those that