1. Concept Overview
Gauss’s law becomes extremely powerful when the charge distribution has symmetry. By choosing a suitable Gaussian surface, the electric field can be calculated quickly without using complicated vector integration. The most common symmetric cases are spherical, cylindrical and planar charge distributions.
2. Electric Field of a Uniformly Charged Sphere
A sphere with uniform charge distribution has perfect spherical symmetry. This allows the field to depend only on the distance from the centre.
2.1. Outside the Sphere (r > R)
The entire charge behaves as if it were concentrated at the centre.
\( E = \dfrac{1}{4 \pi \varepsilon_0} \dfrac{Q}{r^2} \)
The field is identical to that of a point charge.
2.2. Inside the Sphere (r < R)
Only the charge enclosed by the Gaussian surface contributes.
Enclosed charge:
\( Q_{\text{enc}} = Q \dfrac{r^3}{R^3} \)
Electric field:
\( E = \dfrac{1}{4 \pi \varepsilon_0} \dfrac{Q r}{R^3} \)
The field increases linearly with distance from the centre.
3. Electric Field of an Infinite Line Charge
An infinite line of charge has cylindrical symmetry. The field depends only on the distance from the line and is directed radially outward (or inward).
3.1. Gaussian Surface: Cylinder
The field is constant on the curved surface and zero on the ends because it is perpendicular to the end caps.
3.2. Resulting Field
For line charge density \( \lambda \):
\( E = \dfrac{\lambda}{2 \pi \varepsilon_0 r} \)
The field decreases as \( 1/r \).
4. Electric Field of an Infinite Plane Sheet of Charge
An infinite plane sheet has planar symmetry. The electric field is the same everywhere and does not depend on the distance from the sheet.
4.1. Gaussian Surface: Pillbox
A short cylinder (pillbox) is used so that the field passes through its two flat faces and is parallel to its curved surface.
4.2. Resulting Field
For surface charge density \( \sigma \):
\( E = \dfrac{\sigma}{2 \varepsilon_0} \)
This field is uniform and independent of the distance from the plane.
5. Electric Field of a Charged Conducting Shell
A conducting shell forces all excess charge to its outer surface. Inside, the electric field must be zero.
5.1. Outside the Shell
The field is the same as if all charge were at the centre:
\( E = \dfrac{1}{4 \pi \varepsilon_0} \dfrac{Q}{r^2} \)
5.2. Inside the Shell
\( E = 0 \)
This is why conductors shield their interior from external electric fields.
6. Electric Field of a Charged Cylinder (Solid)
A uniformly charged solid cylinder also has cylindrical symmetry. The field behaves differently inside and outside the cylinder.
6.1. Outside the Cylinder
The field depends only on total charge per unit length:
\( E = \dfrac{\lambda}{2 \pi \varepsilon_0 r} \)
6.2. Inside the Cylinder
Enclosed charge varies with cross-sectional area:
\( Q_{\text{enc}} = \lambda \dfrac{r^2}{R^2} \)
Electric field:
\( E = \dfrac{\lambda r}{2 \pi \varepsilon_0 R^2} \)
The field increases linearly with \( r \) inside the cylinder.
7. Using Symmetry to Choose a Gaussian Surface
The correct choice of Gaussian surface is the key to using Gauss’s law effectively.
7.1. Spherical Symmetry
Choose a spherical surface when the charge distribution looks the same in all directions from a point.
7.2. Cylindrical Symmetry
Choose a cylindrical surface when the charge extends infinitely in one direction.
7.3. Planar Symmetry
Choose a pillbox surface when the charge is spread uniformly over a plane.
8. Physical Interpretation
Gauss’s law shows how electric fields “mirror” the symmetry of the charge distribution. When symmetry is high, all points on the Gaussian surface have the same field or have the field aligned in a predictable way. This turns complex calculations into simple expressions, giving electric fields of spheres, cylinders and planes within a few steps.