Patterns of Fields in Space Box versus open surface …no clue… Seem to be able to tell if there are charges inside Gauss’s law: If we know the field distribution on closed surface we can tell what is inside. Define Electric Flux Gauss’s Law Apply it Symmetry

Electric Flux: Direction of E Need a way to quantify pattern of electric field on surface: electric flux 1. Direction flux>0 : electric field comes out flux<0 : electric field goes in +1 -1 Show how total flux becomes zero in the third picture if flux~cos(theta) Relate flux to the angle between outward-going normal and E: flux ~ cos()

Electric Flux: Magnitude of E flux ~ E flux ~ Ecos() 𝑓𝑙𝑢𝑥 ~ 𝐸 ∙ 𝑛 𝑓𝑙𝑢𝑥 ~ 𝐸 ∙ 𝑛 = 𝐸 𝑛 cos⁡(𝜃)

Electric Flux: Surface Area flux through small area: Definition of electric flux on a surface:

Electric Flux: Perpendicular Field or Area Perpendicular area x y q

Adding up the Flux

Gauss’s Law Features: 1. Proportionality constant 2. Size and shape independence 3. Dependence on sum of charges inside 4. Charges outside contribute zero

1. Gauss’s Law: Proportionality Constant What if charge is negative? Works at least for one charge and spherical surface

2. Gauss’s Law: The Size of the Surface universe would be much different if exponent was not exactly 2!

3. Gauss’s Law: The Shape of the Surface ∆ 𝐴 1⊥ ∝ 𝑟 1 2 The flux through the inner sphere is the same as the flux through the outer.

4. Gauss’s Law: Outside Charges – Outside charges contribute 0 to total flux

5. Gauss’s Law: Superposition

Gauss’s Law Features: 1. Proportionality constant 2. Size and shape independence 3. Independence on number of charges inside 4. Charges outside contribute zero Gauss’s Law and Coulomb’s Law? Can derive one from another Last shown. Gauss’s law is more universal: works at relativistic speeds

Applications of Gauss’s Law Knowing E can conclude what is inside Knowing charges inside can conclude what is E

The Electric Field of a Large Plate Symmetry: Field must be perpendicular to surface Eleft=Eright Start here. Could be a sheet of charge or a metal plate with charge Q/A on each side. Assumption: we are finding the field in a region far from the plate edges.

The Electric Field of a Uniform Spherical Shell of Charge Symmetry: Field should be radial The same at every location on spherical surface 𝑟 𝑟 A. Outer Dashed Sphere: B. Inner Dashed Sphere: Finally!

The Electric Field of a Uniform Cube Is Gauss’s law still valid? Yes, it’s always valid. Can we find E using Gauss’s law? Without symmetry, Gauss’s law loses much of its power.

Gauss’s Law for Electric Dipole No symmetry Direction and Magnitude of E varies Numerical Solution Numerical solution to cube with 100 charges per side …

Clicker Question What is the net electric flux through the box? 0 V∙m

Gauss’s Law: Properties of Metal Can we have excess charge inside a metal that is in static equilibrium? Proof by contradiction: =0

Gauss’s Law: Hole in a Metal =0 Less formal: imagine solid piece of metal. remove some (hole) – there are no excess charges, no field – so nothing changes. Is the metal itself as shown electrically neutral? No, apparently, it has a net + charge.

Gauss’s Law: Charges Inside a Hole =0 +5nC

Next Class is a Review