1. GAUSS’ LAW & SYMMETRY Annette Von Jaglinsky Darin Howell Clay Sarafin

2. ELECTRIC FLUX How much an electric field passes through a surface Φ = EA cos Ѳ To total flux, sum all the flux Φ = Σ E dA = We use sigma for finite flux areas and the path integral for infinite flux areas

3. GAUSS’ LAW Amount if flux through a surface is equal to the amount of charge enclosed divided by the vacuum permittivity constant  The distribution of charge does not matter, only the charge does. Common charge density symbols:  λ Lambda = C/m  σ Sigma = C/m^2  ρ Rho = C/m^3 The way to measure this total amount enclosed is by using familiar shapes *To change this symbol into Q multiply by the length/area/volume

4. APPLYING GAUSS: SPHERE Surface area: 4 π r^2 For non conductors:  Non conductors are able to have charge inside the volume so  ---> E(4 π r^2)=Q(Vencl/Vtot)/ ε 0  With this you can solve for E as a function of r For conductors:  All charge collects on the outside surface so the volume enclosed does not matter  ---> E(4 π r^2)=Q/ ε 0  With this you can solve for E as a function of r

5. APPLYING GAUSS: CYLINDERS Surface area is 2 π rl for the body and 2 π r^2 for the end caps When encapsulating a wire/line of charge, use S.A. 2 π rl  ---> E(2 π rl)=Q/ ε 0  Solve for E as a function of r When there is charge on a surface of a conductor use the areas from the end caps of a cylinder  ---> E(2A)= ρ Ah/ ε 0  E = σ /2 ε 0

6. GRAPHS OF CONDUCTING AND NON-CONDUCTING SPHERES Conducting Sphere Non-Conducting Sphere *Where each maximum is kQ/R^2 Decays as 1/ R^2

7. FREE RESPONSE EXAMPLE Two thin, concentric, conducting spherical shells, insulated from each other, have radii of 0.10 m and 0.20 m, as shown here. The inner shell is set at an electric potential of - -100 V, and the outer shell is set at an electric potential of +100 V, with each potential defined relative to the conventional reference point. Let Qi and Qo represent the net charge on the inner and outer shells, respectively, and let r be the radial distance from the center of the shells. Express all algebraic answers in terms of Qi, Qo, r, and fundamental constants, as appropriate. (a) Using Gauss’s Law, derive an algebraic expression for the electric field E(r) for 0.10 m < r < 0.20 m (b) Determine an algebraic expression for the electric field E(r) for r > 0.20 m

8. ANSWER

9. EXAMPLE MC PROBLEM Charge is distributed uniformly throughout a long non-conducting cylinder of radius R. Which of the following graphs best represents the magnitude of the resulting electric field E as a function of r, the distance from the axis of the cylinder?