Overview
This lecture discusses the relationship between Gauss's Law and electric potential around point charges, spherical shells, and conductors, focusing on how the electric field and potential behave in these situations.
Gauss's Law & Point Charges
- Gauss's Law states that the electric flux through a closed surface equals the enclosed charge divided by the permittivity of free space.
- For a point charge, a spherical Gaussian surface has a uniform electric field at distance r: ( E = kQ/r^2 ).
- The electric potential ( V ) outside a point charge is ( V = Q/(4 \pi \epsilon_0 r) ).
- The graph of potential for a point charge follows a ( 1/r ) relationship, spiking near the charge.
Spherical Shells of Charge
- For a charged spherical shell, the potential outside is the same as a point charge: ( V = Q/(4 \pi \epsilon_0 r) ).
- Inside the shell, the electric field is zero and the potential is constant: ( V = Q/(4 \pi \epsilon_0 R) ), where R is the shell's radius.
- The potential graph rises to the shell then remains flat inside.
Solid Conducting Spheres
- In a solid conducting sphere, charge distributes on the surface due to repulsion.
- This creates the same field and potential as a hollow shell: field zero inside, potential constant inside, field and potential outside follow point charge equations.
- The electric potential inside a conductor is constant; the electric field inside is zero.
Field and Potential at the Surface
- At the surface, ( E = Q/(4 \pi \epsilon_0 R^2) ) and ( V = Q/(4 \pi \epsilon_0 R) ).
- Relationship at the surface: ( E = V/R ).
- Smaller radius (sharper curvature) yields higher electric field for the same potential.
- Large electric fields occur at sharp corners, explaining why sparks often jump from them.
Key Terms & Definitions
- Gauss's Law — The total electric flux out of a closed surface is proportional to the enclosed charge.
- Electric Potential (V) — Work done per unit charge to bring a charge from infinity to a point.
- Conductor — A material where electric charges are free to move and reside on the surface.
- Gaussian Surface — An imaginary surface used to apply Gauss’s law.
Action Items / Next Steps
- Review the derivation of electric field and potential for point charges and spherical shells.
- Study the relationship between electric field and potential, focusing on conductors.
- Prepare to explore field-potential relationships in more depth in upcoming lessons.