The mathematical quantity related to number of lines passing through a surface is called the electric flux Æ The electric flux through a surface which is perpendicular to a uniform electric field E is defined as the product of electric field E and surface area A : Æ = EA
Since the electric field is proportional to density of lines of force, the electric flux is proportional to number of lines of force passing through the surface area : Æ aN. If the surface area is not perpendicular to the electric field, then the electric flux is given by
Æ = ®E. n^ A = E cos qA= En A
where n^ is a unit vector perpendicular to the surface and En is the component of electric field perpendicular to the surface (normal component).
The electric flux over a curved surface over which electric field may vary in direction and magnitude can be computed by dividing the surface into large number of very small area elements. Let n^i be the unit vector perpendicular to such an area element and ΔAi be its area. The flux of the electric field through such an area element is Δ Æi = ®Ei. ®ni ΔAi
The total flux through the surface area is found by adding flux through each area element. In the limit ,when number of area elements approaches infinity and area of each element approaches zero, the sum becomes an integral.
Quite often we are interested in finding out the flux through a closed surface. The unit vector n^ in such a case is defined to be directed outward from each point. Note that when an electric line comes out of the closed surface, then ®E.n^ is positive and if it enters the surface ®E. n^ is negative.
The net flux through a closed surface is written as
Ænet = ∫®E. n^ dA = ∫En dA. ….
Please note that Ænet is proportional to the net lines of force going out of the surface, ie, number of lines going out of the surface minus the number of lines going into the surface
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