Ch. 27 : GAUSS’ LAW. Electric Flux. Gauss’ Law. Applications of Gauss’ Law. Conductors. Flux of a Vector Field. rate of flow of a fluid through the surface A: vA ,. vA cosØ (when the rectangle is tilted at an angle Ø.). Flux of a Vector Field.
rate of flow of a fluid through the surface A: vA ,
(when the rectangle is tilted at an angle Ø.)
This rate is defined as the FLUXof the velocity vector v
=> A measure of number of field lines passing through area.
For a flat surface to the field lines
the electric flux
ØE = EA
S.I. unit (Nm2/C)
If surface area is not perpendicular to the field,
A’ = A cos
The # of lines through A’ = the # through A
=> the flux through A’ = the flux through A
ØE = EA cos
magnitude = the area of the ith surface element
direction = perpendicular to the surface element
The flux through the element
DFE = EiDAi cos q = Ei .DAi
the total flux through the surface
= sum of contributions of all elements.
Flux through area element
1 : positive
2 : zero
E along x-axis: To find the net electric flux.
A horizontal uniform field E penetrates the cone.
To find E that enters the left hand side of the cone.
E = - E (1/2 2R h) = - E R h
To find the flux through the netting: relative to the outward normal.
Ans: E = -Ea2
A point charge q is placed at one corner of a cube of edge a .
To find the flux through each of the cube faces:
Ans: E = 0 (for each face touching q);
= 1/240 (for other faces)
This law is useful to calculate E set up by a point charge/collection of charges.
Gauss’ Law relates the net flux through any closed surface to the net charge enclosed by the surface; as:
Spherical Gaussian surfaces
positive point charge
negative point charge.
For the spherical surface,
Flux is independent of the radius r of
the spherical Gaussian surface.
Flux is independent of the size of Gaussian surface
the number of field lines through S1
= number of field lines through S2 or S3.
=> the net flux through any closed surface is independent of the shape of thatsurface
A point charge is located outside closed surface.
The number of lines entering the surface equals the number of lines leaving the surface.
The net flux within the surface is zero.
A spherical gaussian surface surrounds a point charge q. Describe what happens to the total flux through the surface if :
1.the radius of the sphere is doubled
2.the surface is changed to a cube
3. the charge is tripled
4.the charge is moved to another location inside the surface.
The integral form:
When the charge contained within the surface S is continuously distributed within a volume of charge density :
The divergence can be interpreted as the number of field lines starting/terminating at a given point:
=> The number of field lines starting or terminating at a given point is proportional to the charge density at that point.
Electric field at any field point r:
Electric field at a field point r, due to a point charge at origin: