2 4 Conductors capacitance Christopher Crawford PHY 416

§ 2. 4 Conductors – capacitance Christopher Crawford PHY 416 2014 -10 -15

Exam 2 – Friday Oct 24 • Integrate E(r) or V(r) over a charge distribution – Parametrize source points r’(u, …) on surface, path or volume – Calculate field point r and displacement vector, r=r-r’ – Reduce integrals to parameters and constants, including unit vectors • Capacitance calculation – Application of Gauss’ law – Rectangular, cylindrical, or spherical symmetry • Proofs between five formulations of electrostatics – See study sheet • Essay question – prose and diagrams – Relations between field, flux/flow, sources; applied to electrostatics – Geometric interpretation of laws 2

Outline • Conductors vs. dielectrics Charge, field, and potential Induced charge; shielding Electrostatic pressure • Capacitors Field lines, equipotentials Capacitance = flux / flow Capacitance matrix 3

Conductors vs. dielectrics • Conductor – Free vs. bound charge i. metal: conduction band electrons, ~ 1 / atom ii. electrolyte: positive & negative ions • Electrical properties of conductors – Field, potential, charge distribution

Induced charge • Induction in a conductor – displacement of charge – – Charge shifts until electric field is normal to surface Surface charge terminates electric flux lines inside the conductor Total charge remains constant unless there is an escape path 1764 Johan Carl Wilcke invented electrophorus (induction generator) 5

Induction • field lines from charge inside a hollow conductor are communicated outside the conductor by induction – compare: displacement field, Griffiths sections 4. 3, 7. 3 6

Faraday cage • External flux shielded inside a hollow conductor • Consequence of 1/r 2 force law 7

Electrostatic pressure • Force due to electric field on induced charge in conductor – Force per unit area: f = P (or electrostatic pressure) 8

Capacitor • Pair of conductors held at different potential – Electric flux: – Electric flow: • Capacitance: Q = C ΔV – Parallels later in the course: • resistance, reluctance, inductance • Stored energy: E = ½ Q ΔV 9

Example: spherical shells • Two shells of radii a < b 10

Coefficients of capacitance • Linearity of electric potential represented by matrices – Coefficients of potential – Coefficients of capacitance • Two-conductor system 11