Real life Application of Gauss,Green and Stokes Theorem

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