Gauss’s Law AP Physics C

How to use Gauss’s Law Count the lines leaving a surface as + Count the lines entering a surface as – Figures and on p.696

Gauss’s Law Relates the electric field on a closed surface to the net charge within the system For static charges, Gauss’s Law and Coulomb’s Law are EQUIVALENT Gauss’s Law: The net number of lines leaving any surface enclosing the charges is proportional to the net charge enclosed by the surface

Electric Flux Φ The mathematical quantity that corresponds to the number of field lines crossing a surface For a surface perpendicular to the Electric Field, the flux is defined as the product of the magnitude of the field E and the area A: Φ = EA (units are Nm 2 /C)

The box may enclose a charge, by placing a test charge and observing F, we know E. It is only necessary to do this at the surface of the shape.

Pictures of outward (+) flux and inward (-) flux

Electric Flux Φ continued When the area is NOT perpendicular to E, then the following equation is used: Φ = EAcosθ = E n A Where E n is the component of E that is perpendicular or normal to the surface

Flux Flux, in this case Electric Flux, is the amount of (electric) field passing through a specified area. Think of water flowing in a pipe (flux comes from the Latin for “flow”)

Situations where the total flux equals zero

Ф = 0 through triangular prism below. E = 500 N/C 40 cm 50 cm 30 cm 40 cm

The E-field decreases at 1/r 2 while the area increases at r 2 and that increase and decrease cancel each other out and that is why the size of the surface enclosing Q does not matter.

Electric Flux Φ continued What if E varies over a surface? (see Fig on p.697) If we take very small areas A that can be considered a plane, we can then sum the fluxes for each area using Calculus

Flux Symbol  Ф E Unit  Nm 2 /C Equation:

Quantitative Statement of Gauss’s Law P698 The net flux through any surface equals 4πk times the net charge inside the surface

Gauss’ Law

What we can conclude about Ф 1. Ф is proportional to q 2. Whether Ф is inward or outward depends on the q inside the surface 3. A q outside the surface offers zero Ф because Ф in = Ф out

Point Charge

Line of Charge

Sheet of Charge

Uniformly charge insulator at a varying r