PPT: Stokes, Gauss & Green's Theorem | Engineering Mathematics - Civil Engineering (CE) PDF Download

 Page 1


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
Page 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
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).
q
j 
+ve
q
j 
-ve
Page 3


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).
q
j 
+ve
q
j 
-ve
Flux of a Vector Field
? Normal component of vector field transports fluid across 
element of surface area
? Define surface area element as dS = da
1 
x da
2
? Magnitude of normal component of vector field V is 
V.dS = |V||dS| cos(Y)
da
1
da
2
dS
dS = da
1 
x da
2
|dS| = |da
1
| |da
2
|sin(p/2)
Y
dS`
Page 4


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).
q
j 
+ve
q
j 
-ve
Flux of a Vector Field
? Normal component of vector field transports fluid across 
element of surface area
? Define surface area element as dS = da
1 
x da
2
? Magnitude of normal component of vector field V is 
V.dS = |V||dS| cos(Y)
da
1
da
2
dS
dS = da
1 
x da
2
|dS| = |da
1
| |da
2
|sin(p/2)
Y
dS`
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
E
dA
+  +  +  +  +  +
+  +  +  +  +  +
+  +  +  +  +  +
+  +  +  +  +  +
E
dA
Page 5


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).
q
j 
+ve
q
j 
-ve
Flux of a Vector Field
? Normal component of vector field transports fluid across 
element of surface area
? Define surface area element as dS = da
1 
x da
2
? Magnitude of normal component of vector field V is 
V.dS = |V||dS| cos(Y)
da
1
da
2
dS
dS = da
1 
x da
2
|dS| = |da
1
| |da
2
|sin(p/2)
Y
dS`
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
E
dA
+  +  +  +  +  +
+  +  +  +  +  +
+  +  +  +  +  +
+  +  +  +  +  +
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