Gauss’s Law & Electric Flux
Flux of a Vector Field
• Introduction
– Electric field can be determined by Coulomb’s law as
well as by Gauss’s Law
– What is the requirement of Gauss’s Law?
• It gives simpler way of Calculating field
• Gauss’s Law is more General and fundamental
• Coulomb’s Law is valid for static point charges but Gauss’s law
is also valid for rapidly moving charges
– We need to introduce the concept of FLUX (a latin
word)
• Flux of Vector Field – Consider a velocity field of
flowing fluid passing through a closed loop. The flux
  vA
v
A
A
v
A
v
  v( A cos )
• Definition (Number of field lines passing through
an area placed perpendicular to field lines)
• Positive, Negative and zero Flux
• Concept of Source and sink
• Calculation of the flux through the surface
• Generalization of Flux
A5
A2
v
A1
A4
A3
Flux of an Electric Field
• Definition of electric flux:  = E.A
• Flux through an arbitrary surface due to a non
uniform field
• Consider arbitrary closed surface
– Divide the surface in small squares of area ‘dA’
– ‘E’ is then constant on each area
– Consider three different elements
– Negative, positive and zero flux
– Adding up ‘E. dA’ for each element
 E   E.dA
• Electric flux through cylinder
Gauss’s Law
• Gauss’s law relates the total flux through a
closed surface to the net charge ‘q’ enclosed by
the surface
• Step-1: Analyse the collection of positive and
negative charges
• Step-2: Draw an imaginary closed surface
around the collection of charges called Gaussian surface (Important note)
0 E  q
 0  E.dA  q
The circle on the integral sign indicates the closed surface
Example
Electric field of a dipole
S4
+
S1
S3
-
S2
Applications of Gauss’s Law
• Infinite line of Charge
– Step-1: Choose a Gaussian surface (Here cylindrical)
– Step-2: Calculate total charge enclosed
– Step-3: Calculate flux (through cylindrical surface,
upper and lower faces)
– Step-4: Apply Gauss’s law
+
+
+
+
+
+
+
+
+
h
+
+
+
+
+
+
+
+
+
Applications of Gauss’s Law
• Infinite sheet of Charge
• Spherical shell of charge
• Shell Theorems
– A uniform spherical shell of charge behaves, for
external points, as if all the charge was concentrated
at its center
– A uniform spherical shell of charge exerts no
electrical force on a charged particle placed inside
the shell