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

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

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

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()

da1

da2

dS

dS = da1 x da2

|dS| = |da1| |da2|sin(/2)

dS`

Gauss’ Law to charge sheet AND Plate

(C m-3) is the 3D charge density, many applications make use of the 2D density (C m-2):

Uniform sheet of charge density Q/A Same everywhere, outwards on both sides Surface: cylinder sides Inside fields from opposite faces cancel

+ + + + + ++ + + + + +

+ + + + + ++ + + + + +

E

EdA

+ +

+ +

+ +

+ +

+ +

+ +

+ +

+ +

+ +

+ +

+ +

+ +E

dA

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

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.

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.

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.

Thank You