Gauss Law (Theory) & Application To A Point Charge - Field Theory.



- Gauss law is one of the fundamental laws of Electrostatics.


- It states that
“The net electric flux emanating or coming from a close surface S is equal to the total charge contained within the volume V bounded by that surface.”

We know,

Therefore
Ψ = Qenc


Hence Gauss law also states that
 “The total electric flux Ψ through any closed surface is equal to the enclosed electric charge.”


- Charge contained in a volume is given as:
dQ = ρv dv → Q = ∫V ρv dv


Hence we have,



 


Applying Divergence theorem we have,








Comparing the above two equations, we have







And hence 
∇ . D = ρv


This equation is called the 1st Maxwell's equation of electrostatics.


- Gauss law is an easy way of finding electric field for some symmetric problems in electrostatics.


- Gauss law relates the electric field at points on a closed Gaussian surface to the net charge enclosed by that surface.


- One thing to remember is Qenc contains charges which are enclosed within the volume. Charges outside the volume, no matter how large or how close it may be, are not included in the term Qenc.



Procedure To Apply Gauss Law:

- Check out whether symmetric charge distribution exists or not.


- If yes, a hypothetical surface called a Gaussian surface is selected such that E / D (due to charge) is either normal or tangential to the Gaussian surface.


- Gaussian surface is a hypothetical (any imaginary) closed surface enclosing the charge configuration.


- Gaussian surface is a surface to which the electric flux density is normal and over which equal to a constant value.



Application Of Gauss Law To A Point Charge:


- Consider a point charge Q at the origin, say at point P, the electric flux density (D) is to be evaluated.


- Gaussian Surface selected for a point charge is a sphere centered at origin.

From the diagram its clear that


- Electric flux density is everywhere normal to the Gaussian surface i.e. D = Dr ar


- The total charge enclosed by the Gaussian surface, Qenc = Q.


- Since the Gaussian surface is a hollow sphere, hence the variable terms are θ and φ. Thus the differential surface for a hollow sphere is given as:

ds = ∫(φ=0)2π ∫(θ=0)π  (r2 sinθ dθ dφ) ar = 4πr2 ar


- Hence applying gauss law, we have







→ Qenc = Dr 4πr2


Hence

Again we know that, D = ε E.

Therefore




ALSO READ: 

- Gauss's Law - Theory.

- Gauss's Law - Application To a Point charge.

- Gauss's Law - Application To An Infinite Line Charge.

- Gauss's Law - Application To An Infinite Sheet Charge.

- Gauss's Law - Application To a Uniformly Charged Sphere.

- Numericals / Solved Examples - Gauss's Law.

- Scalar Electric Potential / Electrostatic Potential (V).

- Relationship Between Electric Field Intensity (E) and Electrostatic Potential (V).

- Electric Potential Due To a Circular Disk.

- Electric Dipole.

- Numericals / Solved Examples - Electric Potential and Electric Dipole.

- Energy Density In Electrostatic Field / Work Done To Assemble Charges.

- Numericals / Solved Examples - Electrostatic Energy and Energy Density.

- Numericals / Solved Examples - Gauss's law...


- Short Notes/FAQ's

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