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

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?

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: 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

Gauss’s Law: Circuits Can we have excess charge inside in steady state? We already have established that in steady state for a uniform conductor, the E-field is the same magnitude and follows the wire. Thus, Gauss’s law gives no net flux and hence no net charge contained within the conductor. But current carriers are all negative. Where does the positive charge reside?

Gauss’s Law: Junction Between two Wires i1=i2 n1Au1E1 = n2Au2E2 There is negative charge along the interface! n2<n1 u2<u1 Take Gaussian surface just inside the conductor since we are not interested in surface charges. We’ve already shown that within each uniform conductor there is no charge.