Electric Flux and Gauss’s Law

Dr E. Ramanathan PhD

Lecture Video

Lecture Notes: Electric Flux and Gauss’s Law
1. Concept and Definition of Electric Flux
Concept of Electric Flux
Electric flux () represents the number of electric field lines passing through a given surface.
It gives a measure of how much electric field is “flowing” through a surface.
If more field lines pass through a surface, the flux is higher.

Definition of Electric Flux
Mathematically, electric flux is given by:

Where:
= Electric flux (Nm²/C)

= Magnitude of electric field (N/C)

= Area of the surface (m²)

= Angle between the electric field vector and normal to the surface
Key Observations:
If (field perpendicular to surface): is maximum ().
If (field parallel to surface): (no flux).
If the surface is closed, flux measures the net charge enclosed.

2. Gauss’s Law
Statement of Gauss’s Law
Gauss’s Law states that the total electric flux through a closed surface is equal to times the total charge enclosed within the surface.
Mathematical Form:

Where:

= Total electric flux through a closed surface.
= Total charge enclosed within the surface (Coulombs).
= Permittivity of free space ( C²/Nm²).
Derivation of Gauss’s Law
Consider a point charge placed inside a spherical Gaussian surface of radius .
The electric field at every point on the surface is:

The total flux through the sphere is:

Substituting :

Simplifying, we get:

Thus, Gauss’s Law is derived.

3. Applications of Gauss’s Law
A. Electric Field Due to a Uniform Spherical Charge Distribution
Consider a spherical charge distribution of total charge .
By symmetry, the electric field at a distance outside the sphere is:

Key Observations:
Inside a uniformly charged sphere () → Field increases linearly with .
Outside the sphere () → The sphere behaves like a point charge at its center.
B. Electric Field Due to an Infinite Line Charge
Consider an infinite straight line of charge with linear charge density (C/m).
We choose a cylindrical Gaussian surface around the line.
By symmetry, the electric field is radial and same at all points at distance .
Applying Gauss’s Law:

Electric Field Expression:

Key Observation:
The field decreases as (not like point charge).
C. Electric Field Due to an Infinite Plane Sheet of Charge
Consider a large plane sheet of charge with surface charge density (C/m²).
We use a cylindrical Gaussian surface perpendicular to the plane.
By symmetry, the electric field is perpendicular to the sheet.
Applying Gauss’s Law:

Electric Field Expression:

Key Observation:
The field is constant and does not depend on distance (unlike point or line charges).

4. Summary of Key Formulas
Concept
Formula
Key Properties
Electric Flux

Flux depends on field strength and orientation.
Gauss’s Law

Net flux depends only on enclosed charge.
Spherical Charge Distribution

Outside, it behaves like a point charge.
Infinite Line Charge

Field decreases as .
Infinite Plane Sheet of Charge

Field is constant everywhere.

5. Applications of Gauss’s Law
Electrostatic Shielding – Conducting shells prevent external fields from penetrating.
Parallel Plate Capacitors – Store charge and create uniform fields.
Lightning Rods – Work due to strong field concentration at sharp points.
Charged Conductors – Charge distributes evenly on spherical conductors.

Example Problems

🔶 Example 1: Electric Flux through a Tilted Surface

Problem:
A flat surface of area is placed in a uniform electric field of magnitude . The angle between the electric field and the normal to the surface is . Calculate the electric flux through the surface.

✍️ Solution:

Formula:

Given:

✅ Answer:

Explanation:
Flux is maximum when the surface is perpendicular to the field, and reduced by the cosine of the angle when tilted.

🔶 Example 2: Charge Enclosed by a Spherical Surface

Problem:
A spherical Gaussian surface of radius surrounds a point charge. If the electric flux through the surface is , what is the charge enclosed?

✍️ Solution:

Gauss’s Law:

Given:

✅ Answer:

Explanation:
Gauss’s law directly links total electric flux to net enclosed charge. The shape of the surface doesn’t matter—only the total enclosed charge does.

🔶 Example 3: Electric Field of an Infinite Line Charge

Problem:
An infinite line charge has a linear charge density of . Calculate the electric field at a distance of from the line.

✍️ Solution:

Use Gauss’s Law for a line charge:

Given:

✅ Answer:

Explanation:
Gauss’s Law simplifies the process for symmetrical charge distributions. The cylindrical Gaussian surface around the line charge ensures easy calculation.