Gauss's Law Examples

Gauss's Law helps us understand electric fields created by different charge distributions. These examples illustrate how electric fields behave around uniformly charged objects, connecting to key concepts in thermodynamics, electricity, and magnetism in College Physics III.

1. Uniformly charged sphere

   • Electric field inside the sphere is zero; outside, it behaves like a point charge.
   • The electric field at a distance ( r ) from the center (where ( r ) is greater than the radius) is given by ( E = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2} ).
   • Total electric flux through a closed surface is proportional to the enclosed charge.

2. Uniformly charged infinite line

   • Electric field is constant and directed radially outward from the line.
   • The magnitude of the electric field at a distance ( r ) from the line is given by ( E = \frac{\lambda}{2\pi\epsilon_0 r} ), where ( \lambda ) is the linear charge density.
   • The electric field does not depend on the distance along the line.

3. Uniformly charged infinite plane

   • Electric field is uniform and directed perpendicular to the plane.
   • The magnitude of the electric field is given by ( E = \frac{\sigma}{2\epsilon_0} ), where ( \sigma ) is the surface charge density.
   • The electric field is the same on both sides of the plane.

4. Cylindrical shell with uniform charge density

   • Inside the shell, the electric field is zero; outside, it behaves like a line of charge.
   • The electric field at a distance ( r ) from the axis (where ( r ) is greater than the radius) is given by ( E = \frac{\lambda}{2\pi\epsilon_0 r} ).
   • Total electric flux through a closed surface depends on the enclosed charge.

5. Point charge

   • The electric field radiates outward (or inward for negative charges) from the charge.
   • The electric field at a distance ( r ) from the charge is given by ( E = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2} ).
   • The total electric flux through a closed surface surrounding the charge is ( \Phi_E = \frac{Q}{\epsilon_0} ).

6. Spherical shell with uniform surface charge

   • Inside the shell, the electric field is zero; outside, it behaves like a point charge.
   • The electric field at a distance ( r ) from the center (where ( r ) is greater than the radius) is given by ( E = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2} ).
   • The charge distribution affects the electric field only outside the shell.

7. Infinite line of charge

   • Similar to the uniformly charged infinite line, the electric field is constant and directed radially outward.
   • The electric field at a distance ( r ) is given by ( E = \frac{\lambda}{2\pi\epsilon_0 r} ).
   • The field is independent of the length of the line, as it is considered infinite.

8. Uniformly charged solid cylinder

   • Inside the cylinder, the electric field increases linearly with distance from the axis.
   • The electric field at a distance ( r ) from the axis (where ( r ) is less than the radius) is given by ( E = \frac{Qr}{2\pi\epsilon_0 L R^2} ), where ( L ) is the length of the cylinder.
   • Outside the cylinder, it behaves like a point charge.

9. Two parallel infinite planes with equal and opposite charge densities

   • The electric field between the planes is uniform and directed from the positive to the negative plane.
   • The magnitude of the electric field is given by ( E = \frac{\sigma}{\epsilon_0} ), where ( \sigma ) is the charge density of each plane.
   • Outside the planes, the electric field is zero.

10. Uniformly charged thin ring

    • The electric field at points along the axis of the ring is directed along the axis and varies with distance.
    • At the center of the ring, the electric field is zero due to symmetry.
    • The electric field at a distance ( z ) along the axis is given by ( E = \frac{1}{4\pi\epsilon_0} \frac{Qz}{(R^2 + z^2)^{3/2}} ), where ( R ) is the radius of the ring.