Divide shell into rings of charge, each delimited by the angle  and the angle  Use polar coordinates  r  Distance from center: d  r  Rcos  Surface area of ring: Integration    Rcos  Rsin   r d

Chapter 22 Patterns of Fields in Space Electric flux Gauss’s law

What is in the box? no charges? vertical charged plate? Patterns of Fields in Space

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

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 0 Relate flux to the angle between outward-going normal and E: flux ~ cos(  ) Electric Flux: Direction of E

2. Magnitude flux ~ E flux ~ Ecos(  ) Electric Flux: Magnitude of E

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

Perpendicular fieldPerpendicular area xx yy Electric Flux: Perpendicular Field or Area 

Adding up the Flux

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

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

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

All elements of the outer surface can be projected onto corresponding areas on the inner sphere with the same flux 3. Gauss’s Law: The Shape of the Surface

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

5. Gauss’s Law: Superposition

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 Gauss’s law is more universal: works at relativistic speeds Gauss’s Law

1.Knowing E can conclude what is inside 2.Knowing charges inside can conclude what is E Applications of Gauss’s Law

Symmetry: Field must be perpendicular to surface E left =E right The Electric Field of a Large Plate

Symmetry: 1.Field should be radial 2.The same at every location on spherical surface A. Outer Dashed Sphere: B. Inner Dashed Sphere: The Electric Field of a Uniform Spherical Shell of Charge

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

Gauss’s Law for Electric Dipole No symmetry Direction and Magnitude of E varies Numerical Solution

Clicker Question What is the net electric flux through the box?

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

=0 Gauss’s Law: Hole in a Metal

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

Review for Midterm