Gauss' law, Stokes' theorem, and Green's theorem are used to relate line integrals, surface integrals, and volume integrals. Gauss' law relates the electric flux through a closed surface to the enclosed charge. Stokes' theorem converts a line integral around a closed curve into a surface integral over the enclosed surface. Green's theorem converts a line integral around a closed curve into a double integral over the enclosed area. These theorems have applications in electrostatics, electrodynamics, calculating mass and momentum, and deriving Kepler's laws of planetary motion.

1. Real Life Application
of Gauss, Stokes and
Green’s Theorem

2. Gauss’ Law and Applications
 Let E be a simple solid region and S is the boundary surface of E with positive
orientation.Let F be a vector field whose components have continuous partial
derivatives,then
 Coulomb’s Law
 Inverse square law of force
 In superposition, Linear superposition of forces due to all
other charges

3. Electric Field
 Field lines give local direction of field
 Field around positive charge directed
away from charge
 Field around negative charge directed
towards charge
 Principle of superposition used for field
due to a dipole (+ve –ve charge
combination).
qj +ve
qj -ve

4. Flux of a Vector Field
 Normal component of vector field transports fluid across
element of surface area
 Define surface area element as dS = da1 x da2
 Magnitude of normal component of vector field V is
V.dS = |V||dS| cos(Y)
da1
da2
dS
dS = da1 x da2
|dS| = |da1| |da2|sin(p/2)
Y
dS`

5. Gauss’ Law to charge sheet AND
Plate
 r (C m-3) is the 3D charge density, many applications make use
of the 2D density s (C m-2):
 Uniform sheet of charge density s = Q/A
 Same everywhere, outwards on both sides
 Surface: cylinder sides
 Inside fields from opposite faces cancel
+ + + + + +
+ + + + + +
+ + + + + +
+ + + + + +
E
EdA
++++++++++++++++++++++++
E
dA

6. Electrostatic energy of charges
In vacuum
 Potential energy of a pair of point charges
 Potential energy of a group of point charges
 Potential energy of a charge distribution
In a dielectric (later)
 Potential energy of free charges
 Electrostatic energy of charge distribution
 Energy in vacuum in terms

7. Stokes Theorem and
Applications
 Let S be an oriented smooth surface that is bounded
by a simple, closed smooth boundary curve C with
positive orientation. Also let be a vector field then,
 WORK :
- Boundary must be closed
- Transforms closed line integral into surface integral.
 Stokes theorem combined with Gauss’s theorem can
be used for any surface and line integrals.

8. Green’s Theorem and
Applications
 Let C be a positively oriented, piecewise smooth,
simple, closed curve and let D be the region
enclosed by the curve. If P and Q have
continuous first order partial derivatives on
D then,
 Green's Theorem is in fact the special case of
Stokes Theorem in which the surface lies entirely
in the plane.
But with simpler forms. Especially, in a vector field
in the plane.

9. More of greens and Stokes
 In terms of circulation Green's theorem
converts the line integral to a double integral
of the microscopic circulation.
 Water turbines and cyclone may be a
example of stokes and green’s theorem.
 Green’s theorem also used for calculating
mass/area and momenta, to prove kepler’s
law, measuring the energy of steady currents.
Electrodynamics is entirely based on green’s
theorem.

10. Thank You