Direct Energy, 2018a

1 INTRODUCTION 17
The quantity −→ ∇ is called the del operator. In Cartesian coordinates, it
is given by
−→ ∂ ∇ =âx
∂x +â ∂
y
∂y +â ∂
z
∂z . (1.9)
When this operator acts on a scalar function, −→ ∇f, it is called the gradient.
The gradient of a scalar function returns a vector representing the spatial
derivative of the function, and it points in the direction of largest change in
that function. In Maxwell's equations, −→ ∇ acts on vector, instead of scalar,
functions. The operation −→ ∇× −→ E is called the curl, and the operation −→ ∇·−→ E
is called the divergence. Both of these operations represent types of spatial
derivatives of vector functions. The del operator obeys the identity
∇ 2 = −→ ∇·−→ ∇. (1.10)
The operation ∇ 2 f is called the Laplacian of a scalar function, and it
represents the spatial second derivative of that function.
1.6.2 Electromagnetic Waves in Free Space
Electromagnetic waves travel through empty space at the speed of light in
free space, c =2.998 · 10 8 m s , and through other materials at speeds less
than c. For a sinusoidal electromagnetic wave, the speed of propagation is
the product of the frequency and wavelength
| −→ v| = fλ (1.11)
where | −→ v | is the magnitude of the velocity in m s , f is the frequency in Hz,
and λ is the wavelength in meters. In free space, Eq. 1.12 becomes
c = fλ. (1.12)
The speed of light in free space is related to two constants which describe
free space.
c = 1 √
ɛ0 μ 0
(1.13)
The permittivity of free space is given by ɛ 0 = 8.854 · 10 −12 m F where
F represents farads, and the permeability of free space is given by μ 0 =
1.257 · 10 −6 m H where H represents henries. (Constants specied in this
section and in AppendixA are rounded to four signicant digits.)
In free space, the electric eld intensity −→ E and the displacement ux
density −→ D are related by ɛ 0 .
−→ −→ D = ɛ0 E (1.14)