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

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

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

Symmetry makes it simple! 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 Is it a law or a theorem? 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 sphere: B. Inner 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

Gauss’s Law for Electric Dipole

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

=0 What is electric field inside? = 1.No charges on the surface of an empty hole 2.E is zero inside a hole Gauss’s Law: Hole in a Metal

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

Gauss’s Law: Screening Is the field zero inside the box because the metal blocks the field?

Can we have excess charge inside in steady state? Gauss’s Law: Circuits

Gauss’s Law: Junction Between Two Different Metal Wires i 1 =i 2 n 1 Au 1 E 1 = n 2 Au 2 E 2 There is negative charge along the interface! n 2 <n 1 u 2 <u 1

Magnet Cut in Half & Pulled Apart No magnetic monopole! Try to cut a magnet down to a single pole, just get smaller magnets No magnetic Charge!

Dipoles: Electric field: ‘+’ and ‘–’ charges can be separated Magnetic field: no monopoles Suppose magnetic dipole consists of two magnetic monopoles, each producing a magnetic field similar to the electric field. One cannot separate them  total magnetic ‘charge’ is zero. Gauss’s law for magnetism or Gauss’s Law for Magnetism

21.P.22

Patterns of Magnetic Field in Space Is there current passing through these regions? There must be a relationship between the measurements of the magnetic field along a closed path and current flowing through the enclosed area. Ampere’s law

Quantifying the Magnetic Field Pattern Curly character – introduce: Similar to Gauss’s law (Q/  0 )

All the currents in the universe contribute to B but only ones inside the path result in nonzero path integral Ampere’s law is almost equivalent to the Biot-Savart law: but Ampere’s law is relativistically correct Ampère’s Law

Can B have an out of plane component? Is it always parallel to the path? for thick wire: (the same as for thin wire) Would be hard to derive using Biot-Savart law Ampere’s Law: A Long Thick Wire

Number of wires: (N/L)d What is on sides? B outside is very small (solenoid) Uniform: same B no matter where is the path Ampere’s Law: A Solenoid

Symmetry: B || path Is magnetic field constant across the toroid? Ampere’s Law: A Toroid

What is ? A)0 T*m B)8.7e-5 T*m C)1.7 e-4 T*m D)2.0 e-4 T*m E)2.1 e-4 T*m cos(30) =.866

Three equations: Gauss’s law for electricity Gauss’s law for magnetism Ampere’s law for magnetism Is anything missing? ‘Ampere’s law for electricity’ (incomplete) Maxwell’s Equations

Gauss’s law for electricity Gauss’s law for magnetism Incomplete version of Faraday’s law Ampere’s law (Incomplete Ampere-Maxwell law) First two: integrals over a surface Second two: integrals along a path Incomplete: no time dependence Maxwell’s Equations (incomplete)