One of the most significant laws in electrostatics is Gauss’s Law. It clarifies the process by which electric charges produce electric fields. This law’s advantage is that it simplifies the computation of electric fields, particularly in cases when the charge distribution is symmetrical (such as in spheres, cylinders, or planes).
What is Gauss’s Law?
Gauss’s Law states:
“The total electric flux through a closed surface is equal to 1/ε₀ times the total charge enclosed by that surface.”
Mathematically:
ΦE = E ⋅ dA = qin / ε0
Where:
ΦE is the electric flux through the surface,
E is the electric field,
dA is a small vector element of the surface area,
qin is the total charge enclosed inside the surface,
ε0 is the permittivity of free space.
What is Electric Flux?
The amount of electric field that flows through a surface is indicated by its electric flux. Consider spaghetti-like electric field lines passing through a glass (the surface). There is more flux when there are more lines passing through.
If the field is perpendicular to the surface, the flux is maximum.
If it’s parallel, the flux is zero.
Gauss’s Law with an Analogy
Imagine that there are multiple light bulbs (chargers) in a closed room. The number of bulbs inside the room, not outside, determines the flux, or brightness that is visible through the walls. Similarly, when determining the net electric flow, Gauss’s Law states that only the charges inside a closed surface matter.
When and How to Use Gauss’s Law
Gauss’s Law is most useful when there’s symmetry:
Spherical symmetry (e.g., point charge, uniformly charged sphere),
Cylindrical symmetry (e.g., long charged wire),
Planar symmetry (e.g., infinite charged sheet).
Steps to Use Gauss’s Law:
Choose a Gaussian surface that matches the symmetry of the charge distribution.
Calculate the total charge enclosed.
Use the symmetry to simplify the dot product E ⋅ dA
Solve for the electric field.
Examples
Electric Field due to a Point Charge
Choose a spherical surface around the charge.
All points are at the same distance from the charge.
Result:
E = 1 / 4πε0 ⋅ q / r2
(Same result as Coulomb’s Law)
Electric Field due to an Infinite Line of Charge
Use a cylindrical surface around the wire.
Result:
E = λ / 2πε0r
Where λ is the charge per unit length.
Electric Field due to an Infinite Plane Sheet of Charge
Use a “pillbox” (a short cylinder).
Result:
E = σ / 2ε0
Where σ is the surface charge density.
Why Gauss’s Law is Powerful
Simplifies complex problems: Without Gauss’s Law, calculating the electric field for symmetric charge distributions would be much more complicated.
Foundational in electromagnetism: It is one of Maxwell’s four equations that describe the behaviour of electric and magnetic fields.
Teaches about symmetry: Gauss’s Law helps students appreciate the power of symmetry in physics.
Limitations of Gauss’s Law
It’s only easy to apply when there’s symmetry.
It doesn’t tell you the field outside irregularly shaped objects unless you do a full integration.
Important points
Gauss’s law is true for any closed surface, no matter what is its shape of size.
The term q enclosed on the right side of Gauss’s law includes the sum of all charges enclosed by the surface. The charge may be located anywhere in the surface.
Gauss’s law is often useful towards a much easier calculation of the electrostatic field when the system has some symmetry.
Gauss’s law is based on the inverse square dependence on distance contained in the Coulomb’s law. Any violation of Gauss’s law will indicate departure from the inverse square law.
Summary
In electrostatics, Gauss’s Law is a attractively straightforward and effective technique. It offers profound understanding of how charges generate electric fields and facilitates elegant problem-solving in the actual world, particularly in the presence of symmetry.
According to Gauss’s Law, the total charge within a closed surface determines the total electric field that flows through it. It facilitates the calculation of electric fields when the charge distribution and shape are symmetrical.
E ⋅ dA = qin / ε0
This means the electric flux through a closed surface equals the enclosed charge divided by the permittivity of free space (ε0).
When there is symmetry in the charge distribution (spherical, cylindrical, or planar), apply Gauss’s Law. Compared to Coulomb’s Law, it simplifies computations, particularly for continuous charge distributions.
An imaginary closed surface on which we decide to apply Gauss’s Law is called a Gaussian surface. It should be selected in accordance with the charge setup’s symmetry so that the electric field over it may be easily calculated.