Download UNIT 1 : ELECTROSTATICS – LECTURE 5 ELECTRIC FLUX The

UNIT 1 : ELECTROSTATICS – LECTURE 5
ELECTRIC FLUX
The concept of electric flux is useful in association with Gauss' law. The electric flux through a planar
area is defined as the electric field times the component of the area perpendicular to the field. If the area is
not planar, then the evaluation of the flux generally requires an area integral since the angle will be
continually changing.
When the area A is used in a vector operation like this, it is understood that the magnitude of the vector is
equal to the area and the direction of the vector is perpendicular to the area.
Faraday’s experiments showed that a larger positive charge on the inner sphere induced a correspondingly
large negative charge on the outer sphere, which is directly proportional to the electric flux and the charge
on the inner sphere. The constant of proportionality is just unity(=1). Hence the electric flux and the total
charge on the inner sphere are related, by Farady’s Law, as
Ψ=Q
Where the electric flux is also measured in terms of Coulombs.
Electric Flux Density: Electric flux density D is defined as “flux lines per meter square”. It is a vector
with the direction same as flux at the given point, magnitude equal to number of flux lines crossing a
surfac normal to the lines divided by surface area.
Consider the inner sphere as shrinking, i.e., the inner sphere gets smaller and smaller, but retains the same
charge Q. The electric flux density is still the same, given as
Comparing this equation with the equation for electric field,
We can write
D=εE
Where ε is permitivity of the medium, or εo is the permitivity of free space.
GAUSS’S LAW
Statement : The electric flux passing through any cosed surface is equal to the total charge enclosed by
the surface.
Consider the nature of an incremental element of the surface. An incremental element of area ∆S is almost
the portion of a plane surface. And the complete description of this surface requires its magnitude ∆S and
its orientation in space. This direction is normal to the plane which is tangent to the surface at given point.
At point P consider an incremental element of surface ∆S and the flux density component Ds make an
angle θ with ∆S. The flux passing through this surface ∆S can then be written as a product of the surface
area and the flux density component.
The total flux now can be obtained by integrating the above equation over the closed surface
We place a small circle over the integral sign, just to indicate that it is a closed surface. Such surfaces are
often referred to as gaussian surfaces. Hence mathematically,
This can also be written in terms of charge distribution as,
Consider a spherical coordinate system, with point charge Q at origin. Choose a closed surface of
spherical radius r. Now calculating the electric field intensity of the point charge leads to
The differential element of area ona spherical surface in spherical coordinate system is given as,
Consider in cylindrical coordinate system,
The charge enclosed by any surface can be given in cartesian coordinate system as,