Define electric flux. Write its S.I. unit.

“Gauss’s law in electrostatics is true for any closed surface, no matter what its shape or size is”. Justify this statement with the help of a suitable example.

Electric flux is a concept in electromagnetism that quantifies the number of electric field lines passing through a given surface. It essentially measures how much electric field is "flowing" through that surface. Mathematically, electric flux (Φ_E) is defined as the product of the electric field (E) and the area (A) through which it passes, taking into account the angle (θ) between the electric field lines and the normal (perpendicular) to the surface. The formula can be expressed as:

Φ_E = E × A × cos(θ)

The S.I. unit of electric flux is the volt-meter (V·m), which can also be expressed in terms of other fundamental units as newton-meter squared per coulomb (N·m²/C).

Understanding Gauss's Law

Gauss's law states that the total electric flux through a closed surface is directly proportional to the charge enclosed within that surface. This principle holds true regardless of the shape or size of the closed surface. The mathematical expression of Gauss's law is:

Φ_E = Q_enclosed / ε₀

where Q_enclosed is the total charge inside the surface, and ε₀ is the permittivity of free space.

Justifying the Statement

To illustrate why Gauss's law is valid for any closed surface, consider a simple example: a point charge located at the center of a spherical surface. The electric field generated by the point charge radiates outward uniformly in all directions. If we calculate the electric flux through the spherical surface, we can see how Gauss's law applies:

• The electric field (E) at any point on the surface is the same due to symmetry.
• The area (A) of the sphere can be calculated using the formula A = 4πr², where r is the radius of the sphere.
• The angle (θ) between the electric field lines and the normal to the surface is 0 degrees, so cos(θ) = 1.

Substituting these values into the electric flux formula gives:

Φ_E = E × (4πr²) × 1

Now, according to Gauss's law, the total electric flux through the surface should equal the enclosed charge divided by ε₀. If we take a closed surface of any shape—say, a cube or a cylinder—surrounding the same point charge, the same principle applies. The total electric flux through these surfaces will still equal the charge enclosed divided by ε₀, regardless of the surface's geometry.

Real-World Application

This principle is not just theoretical; it has practical applications in various fields, including electrical engineering and physics. For instance, when designing capacitors or understanding electric fields around charged objects, engineers can use Gauss's law to simplify complex calculations, knowing that the shape of the surface does not affect the total flux due to the enclosed charge.

In summary, electric flux provides a way to quantify electric fields through surfaces, and Gauss's law demonstrates that this relationship holds true for any closed surface, making it a powerful tool in electrostatics.