handbook of modern sensors

40 3 Physical Principles of Sensing
For a physicist, any field is a physical quantity that can be specified simultaneously
for all points within a given region of interest. Examples are pressure field, temperature
fields, electric fields, and magnetic fields. A field variable may be a scalar (e.g.,
temperature field) or a vector (e.g., a gravitational field around the Earth). The field
variable may or may not change with time. A vector field may be characterized by
a distribution of vectors which form the so-called flux (). Flux is a convenient
description of many fields, such as electric, magnetic, thermal, and so forth. The
word “flux” is derived from the Latin word fluere (to flow). A familiar analogy of flux
is a stationary, uniform field of fluid flow (water) characterized by a constant flow
vector v, the constant velocity of the fluid at any given point. In case of an electric
field, nothing flows in a formal sense. If we replace v by E (vector representing the
electric field), the field lines form flux. If we imagine a hypothetical closed surface
(Gaussian surface) S, a connection between the charge q and flux can be established as
ε 0 E = q, (3.2)
where ε 0 = 8.8542 × 10 −12 C 2 /N m 2 is the permitivity constant, or by integrating
flux over the surface,
ε 0
∮
E ds = q, (3.3)
where the integral is equal to E . In the above equations, known as Gauss’ law, the
charge q is the net charge surrounded by the Gaussian surface. If a surface encloses
equal and opposite charges, the net flux E is zero. The charge outside the surface
makes no contribution to the value of q, nor does the exact location of the inside
charges affect this value. Gauss’ law can be used to make an important prediction,
namely an exact charge on an insulated conductor is in equilibrium, entirely on its
outer surface. This hypothesis was shown to be true even before either Gauss’ law or
Coulomb’s law was advanced. Coulomb’s law itself can be derived from Gauss’ law.
It states that the force acting on a test charge is inversely proportional to a squared
distance from the charge:
f = 1 qq 0
4πε 0 r 2 . (3.4)
Another result of Gauss’ law is that the electric field outside any spherically symmetrical
distribution of charge (Fig. 3.1B) is directed radially and has magnitude (note
that magnitude is not a vector)
E = 1
4πε 0
q
r 2 , (3.5)
where r is the distance from the sphere’s center.
Similarly, the electric field inside a uniform sphere of charge q is directed radially
and has magnitude
E = 1 qr
4πε 0 R 3 , (3.6)
where R is the sphere’s radius and r is the distance from the sphere’s center. It should
be noted that the electric field in the center of the sphere (r = 0) is equal to zero.