Download Gauss` Law for Electricity

ELECTRO MAGNETIC FIELD WAVES
LECTURE-15
Gauss’ Law for Electricity
Maxwell’s equations can be written either in integral or differential form. Both forms
are equivalent and provide useful insights. Boldface symbols represent vectors that have
components in all three directions.
 D  dS    dv
S
v
(1)
and in differential form,
D  
(2)
In (1) and (2), D is electric flux, S is a vector normal to the surface S, v is some
volume, and  represents space-charge density. In words, the equation states that the only
source that produces a nonzero flux through a closed surface is free electric charge. The
form of (1) makes it clear that the amount of electric flux passing through a closed surface
S is equal to the total charge in the volume enclosed by S. The form of (2) says that the
divergence of the electric flux at a point is equal to the charge density there.
Faraday’s Law
B
 dS
S t
 E  dl  
C
(3)
and in differential form,
E  
B
t
(4)
In (3) and (4), E is the electric field, l is a vector on the contour C, B is the magnetic
flux, and S is a vector normal to the surface S. In words, the equation states that a timevarying magnetic field is a source of an (evidently time-varying) electric field. The form
of (3) makes it clear that a changing flux through a surface S gives rise to an electric field
on the contour of that surface. The form of (4) makes it clear that a changing magnetic
field at a point produces a nonzero curl of E (an E that could produce a current in a small
loop). Faraday’s law is of profound importance since electric generators are based on the
principle of electromagnetic induction.
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Electric flux density:
As stated earlier electric field intensity or simply ‘Electric field' gives the strength of the field at
a particular point. The electric field depends on the material media in which the field is being
considered. The flux density vector is defined to be independent of the material media (as
we'll see that it relates to the charge that is producing it).For a
linear
isotropic medium under consideration; the flux density vector is defined as:
................................................(2.11)
We define the electric flux  as
.....................................(2.12)
Gauss's Law: Gauss's law is one of the fundamental laws of electromagnetism and it states
that the total electric flux through a closed surface is equal to the total charge enclosed by
the surface.
Fig 2.3: Gauss's Law
Let us consider a point charge Q located in an isotropic homogeneous medium of dielectric
constant . The flux density at a distance r on a surface enclosing the charge is given by
...............................................(2.13)
If we consider an elementary area ds, the amount of flux passing through the elementary
area is given by
.....................................(2.14)
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But
, is the elementary solid angle subtended by the area
at the location
of Q. Therefore we can write
For a closed surface enclosing the charge, we can write
which can seen to be same as what we have stated in the definition of Gauss's Law.
Application of Gauss's Law
Gauss's law is particularly useful in computing
or
where the charge distribution has
some symmetry. We shall illustrate the application of Gauss's Law with some examples.
1.An infinite line charge
As the first example of illustration of use of Gauss's law, let consider the problem of
determination of the electric field produced by an infinite line charge of density LC/m. Let us
consider a line charge positioned along the z-axis as shown in Fig. 2.4(a) (next slide). Since
the line charge is assumed to be infinitely long, the electric field will be of the form as shown
in Fig. 2.4(b) (next slide).
If we consider a close cylindrical surface as shown in Fig. 2.4(a), using Gauss's theorm
we can write,
.....................................(2.15)
Considering the fact that the unit normal vector to areas S1 and S3 are perpendicular to the
electric field, the surface integrals for the top and bottom surfaces evaluates to zero. Hence
we can write,
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Fig 2.4: Infinite Line Charge
Maxwell's Equation
Equation (5.1) and (5.2) gives the relationship among the field quantities in the static field.
For time varying case, the relationship among the field vectors written as
(5.20a)
(5.20b)
(5.20c)
(5.20d)
In addition, from the principle of conservation of charges we get the equation of continuity
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(5.21)
The equation 5.20 (a) - (d) must be consistent with equation (5.21).
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