One of the most vital ideas in electrostatics is Gauss’s Law. It aids in our comprehension of how electric charges produce electric fields and how these fields act under various conditions.
Gauss’s Law: The total electric flux through a closed surface is equal to 1 / ε0 times the total charge enclosed by that surface.
Mathematically, E ⋅ dA = qenclosed / ε0
Here,
E is the electric field,
dA is a small area vector on the surface,
ε0 is the permittivity of free space,
and the integral is taken over a closed surface.
Applications of Gauss’s Law
How Gauss’s Law is applied in practical situations to find electric fields.
1.Electric Field Due to an Infinitely Long Charged Wire
Consider a uniformly charged, indefinitely long, straight wire. We select a cylindrical Gaussian surface surrounding the wire in order to determine the electric field at a location close to it.
The electric field is radial (points straight out from the wire) because of symmetry.
The electric field’s strength remains constant across the cylinder’s curved surface.
Using Gauss’s Law, we get:
E ⋅ 2πrL = λL / ε0 ⇒ E = λ / 2πε0r
Where λ is the linear charge density and r is the distance from the wire.
2. Electric Field Due to a Uniformly Charged Spherical Shell
This is like a hollow ball with charge spread evenly on its surface. There are two cases:
(i) Outside the shell (r > R)
Here, we take a spherical Gaussian surface of radius r, greater than the shell’s radius R.
Gauss’s Law gives: E ⋅ 4πr2 = q / ε0 ⇒ E = 1 / 4πε0 ⋅ q / r2
This result is the same as the field due to a point charge placed at the center of the shell.
(ii) Inside the shell (r < R)
In this case, the Gaussian surface encloses no charge (since all the charge is on the shell). E = 0
So, the electric field inside a charged spherical shell is zero.
3. Electric Field Due to a Uniformly Charged Solid Sphere
This is a solid sphere with charge spread throughout its volume. Again, we have two cases:
(i) Outside the sphere (r > R)
Same as the shell case:
E = 1 / 4πε0 ⋅ q / r2
(ii) Inside the sphere (r < R)
Here, only a portion of the total charge is enclosed. If the charge density is ρ, the enclosed charge is: qenclosed = ρ ⋅ 4 / 3πr3
Using Gauss’s Law: E ⋅ 4πr2 = qenclosed / ε0 ⇒ E = ρr / 3ε0
So, the electric field increases linearly with distance inside the sphere.
4. Electric Field Near an Infinite Plane Sheet of Charge
For an infinite plane with uniform surface charge density σ, we use a pillbox-shaped Gaussian surface (like a cylinder with ends parallel to the sheet).
From symmetry:
The field is perpendicular to the surface.
The same field passes through both flat ends.
Applying Gauss’s Law: 2EA = σA / ε0 ⇒ E = σ / 2ε0
The important thing to remember is that the electric field close to an infinite planar sheet is constant and independent of time.
Summary
When symmetry is present, Gauss’s Law is a useful tool that makes calculating electric fields easier. It is particularly helpful for:
Infinite planes (planar symmetry);
Spheres and shells (spherical symmetry);
Long straight wires (cylindrical symmetry).
Gaining an understanding of these applications lays the groundwork for more complex electromagnetism subjects and helps we to understand how electric fields behave in various configurations.
When the charge distribution is symmetrical (spherical, cylindrical, or planar), Gauss’s Law makes calculating electric fields easier. It uses straightforward surface geometry to deliver fast results without requiring complicated integration.
The net charge contained by a Gaussian surface inside a spherical shell is zero. Gauss’s Law states that if there is no enclosed charge, the electric flux is zero, indicating that the electric field is zero throughout the shell.
Although Gauss’s Law is always accurate, it is only helpful in calculations when there is a high degree of symmetry in the charge distribution. The equation still applies to irregular shapes, but it becomes challenging to calculate the field.
Three types of symmetry make Gauss’s Law easy to use:
Spherical symmetry (e.g., point charges, spheres)
Cylindrical symmetry (e.g., long wires)
Planar symmetry (e.g., infinite sheets of charge)