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##### Integration

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**Integration**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: R R Rsin Rcos d r**Chapter 22**Patterns of Fields in Space • Electric flux • Gauss’s law**Patterns of Fields in Space**What is in the box? vertical charged plate? no charges?**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.**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 0 Relate flux to the angle between outward-going normal and E: flux ~ cos()**Electric Flux: Magnitude of E**2. Magnitude flux ~ E flux ~ Ecos()**Electric Flux: Surface Area**Definition of electric flux on a surface: 3. Surface area flux through small area:**Electric Flux: Perpendicular Field or Area**y x Perpendicular field Perpendicular area q**Gauss’s Law**Features: 1. Proportionality constant 2. Size and shape independence 3. Independence on number 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**All elements of the outer surface can be projected onto corresponding areas on the inner sphere with the same flux**4. Gauss’s Law: Outside Charges**– Outside charges contribute 0 to total flux**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 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**The Electric Field of a Uniform Spherical Shell of Charge**• Symmetry: • Field should be radial • The same at every location on spherical surface A. Outer Dashed Sphere: B. Inner Dashed Sphere:**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**Clicker Question**What is the net electric flux through the box? • 0 Vm • 0.36 Vm • 0.84 Vm • 8.04 Vm • 8.52 Vm**Gauss’s Law: Properties of Metal**Can we have excess charge inside a metal that is in static equilibrium? Proof by contradiction: =0