Gauss’s Law Shortcuts and Tricks

1. Electric Field of a Uniformly Charged Sphere

• Remember that the sphere’s internal electric field is zero, and the external field is identical to that of a point charge (Q) concentrated at the sphere’s center.
• For (r) greater than the sphere’s radius (R), use (E=\frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2}). This formula is applicable beyond the boundary of the sphere.
• For (r) less than (R), the electric field is zero because the enclosed charge within this radius is zero.

2. Electric Field of an Infinite Charged Plane

• The electric field has a constant magnitude (E=\frac{\sigma}{2\varepsilon_0}) that points away from the plane. This field is equal on both sides of the plane.
• Remember that the field strength is independent of the distance (d) from the plane.
• This shortcut applies only to infinite planes, not finite ones.

3. Electric Field between Parallel Charged Plates

• The field is constant between the plates and zero everywhere else.
• The field direction is from the higher potential plate to the lower potential plate.
• The magnitude is given by (E = \frac{\sigma}{\varepsilon_0}) where (\sigma) is the surface charge density of either plate and (\varepsilon_0) is the permittivity of free space.

4. Electric Flux through a Closed Surface

• Gauss’s law states that the total electric flux through any closed surface is equal to the net charge enclosed by the surface divided by the permittivity of free space, regardless of the shape or size of the surface.
• If there is a net positive charge enclosed, then the electric flux will be positive. If the net charge is negative, then the flux will be negative.
• If there is no net charge enclosed, then the electric flux will be zero.