Last time… Fields, forces, work, and potential Electric forces and work + + Electric potential energy and electric potential Tues. Oct. 7, 2008 Physics 208 Lecture 11

Potential from electric field V=Vo Potential change is (1/q) * (work done to move positive charge from A to B) Positive work required in direction opposite to field, negative work required in same direction as field. Ask student what will the answer be for each direction as you go through slide. dV largest in direction of E-field. dV smallest (zero) perpendicular to E-field Tues. Oct. 7, 2008 Physics 208 Lecture 11

Equipotential lines Lines of constant potential Mention lab this week is tracing out equipotential contours and electric field vectors. Lines of constant potential In 3D, surfaces of constant potential Tues. Oct. 7, 2008 Physics 208 Lecture 11

Topographic map Each lines is constant elevation Same as constant gravitational potential gh (energy = mgh) Height interval between lines constant Tues. Oct. 7, 2008 Physics 208 Lecture 11

Electric field from potential Said before that Spell out the vectors: This works for Argument is to consider displacements only along dx, only along dy, etc. Can then identify Ex=dV/dx. Must hold for all general displacements. Usually written Tues. Oct. 7, 2008 Physics 208 Lecture 11

Quick Quiz Suppose the electric potential is constant everywhere. What is the electric field? Positive Negative Increasing Decreasing Zero Tues. Oct. 7, 2008 Physics 208 Lecture 11

Electric Potential - Uniform Field B Constant E-field corresponds to linearly decreasing (in direction of E) potential A Here V depends only on x, not on y x Tues. Oct. 7, 2008 Physics 208 Lecture 11

Check of basic cases Previous quick quiz: uniform potential corresponds to zero electric field Linear potential corresponds to constant electric field Tues. Oct. 7, 2008 Physics 208 Lecture 11

Potential ( V ) of spherical conductor What is V of spherical conductor relative to infinity? Charge on surface  spherical charge shell Gauss’ law  E = keQ / r2 in the radial direction V is work / Coulomb to bring point charge from ∞ Comment that equipotentials look like that for a point charge. But what is the actual value? Tues. Oct. 7, 2008 Physics 208 Lecture 11

Quick quiz Previous result says conducting sphere of radius R carrying charge Q is at a potential Two conducting spheres of diff radii connected by long conducting wire. What is approximately true of Q1, Q2? A) Q2>Q1 B) Q2<Q1 C) Q2=Q1 R1 R2 Q1 Q2 Tues. Oct. 7, 2008 Physics 208 Lecture 11

Connected spheres Since both must be at the same potential, Smaller radius sphere has smaller charge Surface charge densities? Smaller sphere has larger surface charge density Electric field? Local E-field bigger at more sharply curved (smaller R) regions Tues. Oct. 7, 2008 Physics 208 Lecture 11

Varying E-fields on conductor Now do demos: Shaped conductor, paddle, faraday bucket, electric pinwheel, lightning rod and ball. Larger electric fields near smaller radii surfaces. Large electric fields at sharp points, Strong fields can ionize air atoms. Tues. Oct. 7, 2008 Physics 208 Lecture 11

Potential and charge Have shown that a conductor has an electric potential, and that potential depends on its charge For a charged conducting sphere: + + + + + + + + Electric potential proportional to total charge + + + Tues. Oct. 7, 2008 Physics 208 Lecture 11

Quick Quiz Consider this conducting object. When it has total charge Qo, its electric potential is Vo. When it has charge 2Qo, its electric potential A. is Vo B. is 2Vo C. is 4Vo D. depends on shape Key is that charge will distribute in exactly the same way, independent of how much charge is there. Tues. Oct. 7, 2008 Physics 208 Lecture 11

Capacitance Electric potential of any conducting object proportional to its total charge. C = capacitance Large capacitance: need lots of charge to change potential Small capacitance: small charge can change potential. Tues. Oct. 7, 2008 Physics 208 Lecture 11

Capacitors Where did the charge come from? Usually transferred from another conducting object, leaving opposite charge behind A capacitor consists of two conductors Conductors generically called ‘plates’ Charge transferred between plates Plates carry equal and opposite charges Potential difference between plates proportional to charge transferred Q Tues. Oct. 7, 2008 Physics 208 Lecture 11

Definition of Capacitance Same as for single conductor but V = potential difference between plates Q = charge transferred between plates SI unit of capacitance is farad (F) = 1 Coulomb / Volt This is a very large unit: typically use mF = 10-6 F, nF = 10-9 F, pF = 10-12 F Tues. Oct. 7, 2008 Physics 208 Lecture 11

How was charge transferred? Battery has fixed electric potential difference across its terminals Conducting plates connected to battery terminals by conducting wires. DVplates = DVbattery across plates Electrons move from negative battery terminal to -Q plate from +Q plate to positive battery terminal This charge motion requires work The battery supplies the work DV Do demo of blowing up aluminum foil with capacitor. Calculate total charge on capacitor. Tues. Oct. 7, 2008 Physics 208 Lecture 11

Work done to charge a capacitor Requires work to transfer charge dq from one plate: Total work = sum of incremental work Comment that potential changes as charge is transferred between plates -> integrate Work done stored as potential energy in capacitor Tues. Oct. 7, 2008 Physics 208 Lecture 11

Example: Parallel plate capacitor +Q -Q d inner outer Charge Q moved from right conductor to left conductor Charge only on inner surfaces Plate surfaces are charge sheets, each producing E-field Uniform field between plates Tues. Oct. 7, 2008 Physics 208 Lecture 11

Quick Quiz Electric field between plates of infinite parallel-plate capacitor has a constant value /o. What is the field outside of the plates? /o /2o - /2o /4o Tues. Oct. 7, 2008 Physics 208 Lecture 11

What is potential difference? + - Potential difference = V+-V- = - (work to move charge q        from + plate to - plate) / q Charge only on inner surfaces means that these are sheets of charge. Or can use Gauss’ law to find same electric field. d +Q -Q Tues. Oct. 7, 2008 Physics 208 Lecture 11

What is the capacitance? -Q This is a geometrical factor +Q Energy stored in parallel-plate capacitor Energy density d Tues. Oct. 7, 2008 Physics 208 Lecture 11