Gauss law in electrostatics
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Electrostatics investigates interaction between fixed electric charges
The Gauss' Law is used to find electric field when the charge is continuously distributed within an object with symmetrical geometry, such as sphere, cylinder, or plane. Gauss' law follows Coulomb's law and the Superposition Principle.
Electric flux
Consider electric field and choose mentally a plane surface of area A with normal unit vector (with magnitude u = 1), oriented at angle with respect to the field as shown below
Electric flux quantities
Let the area be so small that the electric field is uniform everywhere on it.
The electric flux thorough the area A is defined as scalar product
The above definition defines the units of electric flux as volt times meter
Now we can define the electric flux through any surface in an arbitrary electric field. We should subdivide this surface into n small elements each of area , and find the electric fluxes through all elements using the above formula
, for
where: is the electric field at the center of ; is the angle between and normal to the plane of the element.
Since electric flux is a scalar value, the total electric flux through the surface is given by the sum
Here the smaller value the more n, and the more accurate value of the electric flux is given.
Two important cases follows from the above formula:
1. When the electric field everywhere is parallel to the surface the electric flux through the surface is zero, since here and .
2. When the electric field is perpendicular everywhere to the surface the electric flux through the surface is given by
In particular, if the electric field everywhere is normal to the surface and has the same value E, then the formula for the electric flux is:
where A is the total area of the surface
Using the electric field lines concept, let us find the obvious physical meaning of the electric flux. Putting and in formula for electric flux, we get the number of electric field lines crossing the element of area
Now putting it into electric flux, we get
Thus the net electric flux through any surface is equal numerically to the total number of electric lines, N, crossing the surface.
Now consider the general case when the electric field is produced by a system of point charges Qi for i = 1,2,3,...n. Draw mentally a closed surface with area A around the charges so they all are enclosed within the surface. Such surface is called Gauss' surface.
Gauss' surfaUsing this symmetry, insert Gauss’ surface as a closed cylinder with cross section A. The electric field everywhere on the lateral surface is parallel to the surface, and so according to case 1 the electric flux through the lateral surface is zero. On the faceplates the electric field is normal to the surface and has a constant value E, so according to case 2 the net electric flux through the cylinder is given by
According to the definition of surface charge density, the charge enclosed within the cylinder is given
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