PPT No. 3
* Divergence Theorem
* Gauss’s law- Integral form
* Gauss’s law- Differential Form
Gauss’ Law
Gauss’s Law- Integral Form
Statement of the Gauss’s Law
Electric Flux through any closed surface
enclosing a charge QS is
proportional to the enclosed electric charge
Gauss’ Law
Quantitatively Gauss’s Law is given by
for Charge QS inside the surface
=0
for Charges outside the surface
QS is the total charge enclosed by the surface
(including both Free and Bound charge) and
ε0 is the permittivity of free space or Electric constant
Gauss’ Law
LHS is a surface integral of E-Field through
a closed surface S denoting the electric flux ΦE
RHS is the total charge Qs enclosed by S
divided by the Electric constant ε0
S is called as Gaussian surface
Only the charge enclosed within a volume
defined by a closed surface contributes to
the net electric flux through the surface
Gauss’ Law- Significance
Gauss's law relates
the distribution of electric charge Qs
to the resulting electric field E.
Gauss Law is
One of the fundamental laws of physics
Gauss’ Law is
One of the four Maxwell’s Equations
which form the basis of Classical Electrodynamics
Gauss's law can be used to derive Coulomb’s Law &
Coulomb’s Law can be used to derive Gauss' Law
Gauss’ Law- Applications
Gauss's law is very useful
to find the distribution of electric charge
when the electric field E is known
By integrating the electric field,
the flux can be found and
the charge distribution in the region
can be deduced by using Gauss Law.
Gauss's law can be employed in the reverse problem also
(i.e. The electric charge distribution is known and
the electric field needs to be computed) under certain conditions
Types of Charge Distributions
•3-D Charge distributions are specified in terms of
• Volume charge density ρ defined as
•Total Charge / Volume
•2-D Charge distributions are specified in
•Surface charge density σ defined as
•Total Charge / Area
•1-D Charge distributions are specified in terms of
• Linear charge density λ defined as
•Total Charge / Length
Gauss’s Law
If the electric charge distribution has some kind symmetry,
then the electric field E passes through the surface uniformly and
E can be computed using Gauss Law e. g.
The Gaussian surface is
a hypothetical surface enclosing the charge
constucted to calculate the flux or electric field
Selection of the Gaussian surface
Selection of the proper Gaussian surface is
the Key
to solve the problems using Gauss Law
Usually the shape of the Gaussian surface is such that
it mimics the symmetry of the charge distribution
The Gaussian surface is constructed so that
the electric flux density is normal to the surface and
it has a constant value on the surface.
Selection of the Gaussian surface
Charge Distributions having Spherical Symmetry are
* A point charge
A charged sphere
Charged spherical shell
Their Gaussian Surface is
Concentric sphere with centre at the charge
Selection of the Gaussian surface
Charge Distributions having Cylindrical Symmetry are
Long straight line of charge
Charged infinite cylinder
Their Gaussian Surface is
A Concentric cylinder
with axis along the line of charge
Selection of the Gaussian surface
Charge Distributions having Planar Symmetry are
Infinite layer of charge
Infinite plane sheet
Their Gaussian Surface is
A Cylinder with sides ┴ to sheet i.e.
Faces ║ to sheet
Selection of the Gaussian surface
A coordinate system is selected that
most nearly matches the symmetry of the charge distribution
Spherical co-ordinate system is used to determine
the flux due to a point charge
as its flux has spherical symmetry
Cartesian co-ordinate system is used to determine
the flux due to an Infinite plane sheet of charge
as charge-sheet along x-y plane has
E-field along z-axis which is independent of x or y axis &
symmetric under reflection about the x-y plane
due to its planar symmetry
Gauss' Law
If symmetry of any kind is not present
in electric charge distribution,
then it is difficult to compute electric field E
from the given distribution of electric charge Q
using Gauss’s law in Integral form
In that case
the differential form of Gauss's law is useful
Gauss’ Law- Differential Form
According to the integral form of Gauss's law,
where charge Q is in terms of Volume charge density ρ,
dτ is an infinitesimal element of volume.
Using the Divergence theorem
LHS is a volume integral over the volume V,
RHS is the surface integral over the boundary of volume V.
Gauss’ Law- Differential Form (contd)
where
· E is the Divergence of the electric field
ρ is the Volume Charge density
ε0 is the electric constant.
This is the differential form of Gauss's law
Equivalence of Two Forms of Gauss's law
Integral form of Gauss's law
Differential form of Gauss's law
Equivalence of Two Forms of Gauss's law
The Integral and Differential forms of Gauss's law
are equivalent and
are related by the Divergence theorem.