Short Answer:

Gauss’s Law states that the total electric flux through a closed surface is directly proportional to the total electric charge enclosed within that surface. It is a basic law in electrostatics that helps to calculate electric fields using the concept of symmetry. The electric field lines start or end only on electric charges, and Gauss’s Law relates these lines to the charge inside a closed surface.

Mathematically, Gauss’s Law is written as:

ΦE=qinsideε0\Phi_E = \frac{q_{\text{inside}}}{\varepsilon_0}ΦE​=ε0​qinside​​

Where ΦE\Phi_EΦE​ is the total electric flux, qinsideq_{\text{inside}}qinside​ is the total enclosed charge, and ε0\varepsilon_0ε0​ is the permittivity of free space.

Detailed Explanation:

Gauss’s Law

Gauss’s Law is one of the four Maxwell’s equations and is a fundamental law in the study of electric fields. It was proposed by Carl Friedrich Gauss, a German mathematician and physicist. The law provides a simple way to calculate electric fields when there is symmetry in the system, such as spherical, cylindrical, or planar symmetry.

The law connects the electric flux passing through a closed surface with the electric charge enclosed by that surface. The surface used in Gauss’s Law is called a Gaussian surface, and it can be any imaginary closed shape, like a sphere, cube, or cylinder.

Statement of Gauss’s Law

“The total electric flux through a closed surface is equal to 1ε0\frac{1}{\varepsilon_0}ε0​1​ times the total electric charge enclosed within the surface.”

This means that the electric field coming out of a closed surface is only due to the charges inside that surface. Charges outside the surface do not contribute to the net flux through it.

Mathematical form

ΦE=∮E⃗⋅dA⃗=qinsideε0\Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{q_{\text{inside}}}{\varepsilon_0}ΦE​=∮E⋅dA=ε0​qinside​​

Where:

• ΦE\Phi_EΦE​ = total electric flux
• E⃗\vec{E}E = electric field
• dA⃗d\vec{A}dA = small area vector of the surface
• qinsideq_{\text{inside}}qinside​ = total charge enclosed by the surface
• ε0\varepsilon_0ε0​ = permittivity of free space (8.85×10−12 C2/N\cdotpm2)(8.85 \times 10^{-12} \, \text{C}^2/\text{N·m}^2)(8.85×10−12C2/N\cdotpm2)

The dot product E⃗⋅dA⃗\vec{E} \cdot d\vec{A}E⋅dA shows that only the component of the electric field perpendicular to the surface contributes to the flux.

Applications of Gauss’s Law

1. Spherical symmetry:
   Useful for finding the electric field around point charges, charged spheres, and shells.
2. Cylindrical symmetry:
   Helps calculate the field near long charged wires.
3. Planar symmetry:
   Used for large flat sheets of charge.
4. Inside conductors:
   Gauss’s Law proves that the electric field inside a conductor is zero in electrostatic conditions.
5. Design of capacitors:
   Helps in analyzing field distribution between plates.

Important points

• The law is valid for any closed surface, whether physical or imaginary.
• It works with any charge distribution, but is most useful in cases with symmetry.
• Gauss’s Law is equivalent to Coulomb’s Law, but more powerful in solving field problems with complex geometry.
• It also confirms that electric field lines begin and end on charges.

Real-life example

Consider a uniformly charged metal sphere. Using Gauss’s Law with a spherical Gaussian surface, we can easily calculate the electric field outside the sphere without complicated integration. This simple technique is widely used in physics and electrical engineering.

Conclusion:

Gauss’s Law relates the electric flux through a closed surface to the charge enclosed by that surface. It simplifies the calculation of electric fields in symmetric situations and helps understand the behavior of fields in and around charged bodies. It is a powerful and essential tool in the study of electrostatics.