PHYS 208 Lecture 5

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Gauss's Law

Symmetric disributions

Spherically symmetric distributions can be treated as a single point charge located at the center

Infinitely long cylindrical distributions (or lines) have symmetry and E-fields are radially outward from every point

Infinite planes/sheets also have fields that are radially outward from every point

Example

A uniformly charhed insulating sphere of radius R with chorge density ρ. Find the flux for r < R and r > R:

For r < R, only the enclosed Gaussian spherical surface should be included:

$q=\int \rho \,\mathrm {d} v={\frac {4}{3}}\pi r^{3}\rho$

$\Phi =EA=E(4\pi r^{2})={\frac {q}{\epsilon _{0}}}$ $E(r)={\frac {r\rho }{3\epsilon _{0}}}$

For r > R, the entire sphere acts like a point charge, where $q=\rho v={\frac {4}{3}}\pi r^{3}\rho$

$E={\frac {q}{4\pi \epsilon _{0}r^{2}}}$

Conductors

We are dealing with electrostatics in equilibrium: the charges do not move and the net force is zero.

Electric field inside a conductive material is zero

If we put a charge inside a hole in the conductor, there is a field inside the hole. The charge inside the conductor has to be 0, so an induced charge must have taken place: opposite charges must be on the surface of the hole (enough to counter the charge inside the hole). The charge inside the hole also adds its charge to the charge along the outer surface