Magnetic Forces in Moving Reference Frames Electric force: Two protons +e r 1 2 v Magnetic field: F21,m B1 Magnetic force: F21,e E1 Coulomb force on moving charges: approximation, E is not exactly the same (address later). Recall that we have already said that two current carrying wires experience an attractive magnetic force if current direction is the same.

Magnetic Forces in Moving Reference Frames Electric force: +e r 1 2 v E1 F21,e B1 F21,m Magnetic force: Ratio: Coulomb force on moving charges: approximation, E is not exactly the same (address later) =c2 (m/s)2 it is not accidental!

Magnetic Forces in Moving Reference Frames 1 2 v E1 F21,e B1 F21,m For v<<c the magnetic force is much smaller than electric force Ask students to reconcile this with the fact that two wires carrying current in the same direction are attracted to each other. Full Lorentz force: downward

Magnetic Forces in Moving Reference Frames 20 ns +e r 1 2 v E1 F21,e B1 F21,m 15 ns Who will see protons hit floor and ceiling first? This is a kind of phenomenon which led Einstein to develop theory of relativity

Relativistic Field Transformations Our detailed derivations are not correct for relativistic speeds, but the ratio Fm/Fe is the same for any speed: According to the theory of relativity:

Magnetic Field of a Moving Particle Still: Moving: Slow case: v<<c  Field transformation is consistent with Biot-Savart law What about B below particle? Electric and magnetic fields are interrelated Magnetic fields are relativistic consequence of electric fields

Electric Field of a Rapidly Moving Particle

The Principle of Relativity There may be different mechanisms for different observers in different reference frames, but all observers can correctly predict what will happen in their own frames, using the same relativistically correct physical laws.

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

Patterns of Fields in Space What is in the box? no charges? vertical charged plate? Gauss’s law: If we know the field distribution on closed surface we can tell what is inside.

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

Adding up the Flux

Clicker Question What is the electric flux through the area A? E = 100 V/m q = 30o DA = 2 m2 100 V*m 173 V*m 50 V*m 87 V*m c

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 For negative charge cos is negative 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

5. Gauss’s Law: Superposition

Gauss’s Law Is it a law or a theorem? Can derive one from another Last shown. Gauss’s law is more universal: works at relativistic speeds

Clicker Question What is the net electric flux on the box? 0 V*m

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 A. Outer sphere: B. Inner sphere:

The Electric Field of a Uniform Cube Is Gauss’s law still valid? Can we find E using Gauss’s law?

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 What is electric field inside the hole? = 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. No charges on the surface of an empty hole E is zero inside a hole

Gauss’s Law: Screening Similar to a hole in the metal

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