Transcript ppt
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 1 Potential from electric field dV E d V Vo d d E d V Vo E d V=Vo V Vo E d dV largest in direction of E-field. dV smallest (zero) perpendicular to E-field Tues. Oct. 7, 2008 Physics 208 Lecture 11 2 Equipotential lines Lines of constant potential In 3D, surfaces of constant potential Tues. Oct. 7, 2008 Physics 208 Lecture 11 3 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 4 Electric field from potential Said before that dW Fext ds FCoulomb ds dV E ds Spell out the vectors: for This works dV E x dx E y dy E z dz dV dV dV Ex , Ey , Ez dx dy dz Usually written Tues. Oct. 7, 2008 dV dV dV E V , , dx dy dz Physics 208 Lecture 11 5 Quick Quiz Suppose the electric potential is constant everywhere. What is the electric field? A) Positive B) Negative C) Increasing D) Decreasing E) Zero Tues. Oct. 7, 2008 Physics 208 Lecture 11 6 Electric Potential - Uniform Field dV E ds VB VA B B B E ds Exˆ ds A Edx E A B A dx E x B xA A B A Constant E-field corresponds to linearly decreasing (in direction of E) potential Here V depends only on x, not on y x Tues. Oct. 7, 2008 Physics 208 Lecture 11 7 Check of basic cases Previous quick quiz: uniform potential corresponds to zero electric field E V constant 0 Linear potential corresponds to constant electric field E V Ex Ex, Ex, Ex Exˆ y z x Tues. Oct. 7, 2008 Physics 208 Lecture 11 8 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 ∞ V R V E R E ds R ds Q E dr k 2 dr r R Q Q k k rR R Tues. Oct. 7, 2008 E ds E dr Edr Physics 208 Lecture 11 9 Quick quiz Previous result says conducting sphere of radius R carrying charge Q is at a potential kQ/ R Two conducting spheres of diff radii connected by long conducting wire. What is approximately true of Q1, Q2? R1 Q1 Q2 R2 A) Q2>Q1 B) Q2<Q1 C) Q2=Q1 Tues. Oct. 7, 2008 Physics 208 Lecture 11 10 Connected spheres Since both must be at the same potential, kQ1 kQ2 R2 Q2 Q1 R1 R2 R1 Smaller radius sphere has smaller charge Surface charge densities? Q R1 1 2 2 4R R2 Electric field? R E E 2 1 1 o R2 Tues. Oct. 7, 2008 Smaller sphere has larger surface charge density Local E-field bigger at more sharply curved (smaller R) regions Physics 208 Lecture 11 11 Varying E-fields on conductor 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 12 Potential and charge Have shown that a conductor has an electric potential, and that potential depends on its charge For a charged conducting sphere: + + + + + + + + + + + Tues. Oct. 7, 2008 Q k V R V k Q R R Electric potential proportional to total charge Physics 208 Lecture 11 13 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 Tues. Oct. 7, 2008 Physics 208 Lecture 11 14 Capacitance Electric potential of any conducting object proportional to its total charge. 1 V Q C 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 15 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 16 Definition of Capacitance Same as for single conductor 1 V Q C 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 17 How was charge transferred? Battery has fixed electric potential difference across its terminals Conducting plates connected to battery terminals by conducting wires. Vplates = Vbattery across plates Electrons move V from negative battery terminal to -Q plate from +Q plate to positive battery terminal This charge motion requires work The battery supplies the work Q CV Tues. Oct. 7, 2008 Physics 208 Lecture 11 18 Work done to charge a capacitor Requires work to transfer charge dq from one plate: q dW Vdq dq C Total work = sum of incremental work Q W 0 q Q2 dq C 2C Work done stored as potential energy in capacitor Q2 1 1 2 U QV CV 2C 2 2 Tues. Oct. 7, 2008 Physics 208 Lecture 11 19 Example: Parallel plate capacitor +Q Charge Q moved from right outer conductor to left conductor -Q inner Charge only on inner surfaces Plate surfaces are charge sheets, each producing E-field E left E right /2o /2o /o Uniform field between plates d Tues. Oct. 7, 2008 Physics 208 Lecture 11 20 Quick Quiz Electric field between plates of infinite parallel-plate capacitor has a constant value /o. What is the field outside of the plates? A. /o B. /2o C. - /2o D. /4o E. 0 Tues. Oct. 7, 2008 Physics 208 Lecture 11 21 What is potential difference? Potential difference = V+-V= - (work to move charge q from + plate to plate) / q qEd/q d V Ed d /o Q o A d V V V Q o A Tues. Oct. 7, 2008 + + + + + + + + + + + + + + + +Q Physics 208 Lecture 11 E d - -Q 22 What is the capacitance? d V V V Q o A V Q /C C o A d -Q This is a geometrical factor +Q Energy stored in parallel-plate capacitor 1 1 o A 1 2 2 U CV Ed Ado E 2 2 2 d Energy density Tues. Oct. 7, 2008 2 1 U /Ad o E 2 2 Physics 208 Lecture 11 d 23