Electricity and Magnetism--Conductors and Capacitors

1 8.022 (E&M) – Lecture 5 Topics: 􀂄 More on conductors… and many demos! 􀂄 Capacitors G. Sciolla – MIT 8.022 – Lecture 5 2 Last time… 􀂄 Curl: 􀂄 Stoke’s theorem: 􀂄 Laplacian: 􀂄 Conductors 􀂄 Materials with free electrons (e.g. metals) 􀂄 Properties: 􀂄 Inside a conductor E=0 􀂄 Esurface=4πσ 􀂄 Field lines perpendicular to the surface 􀃆 surface is equipotential 􀂄 Uniqueness Theorem 􀂄 Given ρ(xyz) and boundary conditions, the solution φ(xyz) is unique A = curl 0 CF ds F dA E ⇒ ∇× = ∫ ∫ 􀁇 􀁇 􀁇 􀁇 􀁇 i i 􀁶 curl F F = ∇ × 􀁇 􀁇 2 φ φ ∇ ≡ ∇ ∇ i in vacuum 2 2 4 (Poisson) 0 (Laplace) φ πρ φ ∇ = − ⎯⎯⎯⎯→∇ =2 G. Sciolla – MIT 8.022 – Lecture 5 3 Charge distribution on a conductor 􀂄 Let’s deposit a charge Q on a tear drop-shaped conductor 􀂄 How will the charge distribute on the surface? Uniformly? + ++ + + + + + + + + ? 􀂄 Experimental answer: NO! (Demo D28) 􀂄 σtip >> σflat 􀂄 Important consequence 􀂄 Although φ=const, E=4πσ 􀃆 Etip >> Eflat 􀂄 Why? + + + + + + + + + ++ ! G. Sciolla – MIT 8.022 – Lecture 5 4 Charge distribution on a conductor (2) 􀂄 Qualitative explanation 􀂄 Consider 2 spherical conductors connected by conductive wire 􀂄 Radii: R1 and R2 with R1 >> R2 􀂄 Deposit a charge Q on one of them 􀃆 charge redistributes itself until φ=constant R1 R2 1 2 1 2 1 2 1 1 1 2 1 2 1 211 2 1 2 1 2 2 2 22 2 E = E E =Q Q R R Q E R R R R R R QR R φ φ φ σσ φ ⎧ = = = ⎪⎪⎪= ⇒ = ⇒ = ⎨⎪⎪= ⎪⎩ 􀂄 Conclusion: Electric field is stronger where curvature (1/R) is larger 􀂄 More experimental evidence: D29 (Lightning with Van der Graaf)3 G. Sciolla – MIT 8.022 – Lecture 5 5 Shielding We proved that in a hollow region inside a conductor E=0 􀂄 This is the principle of shielding 􀂄 Do we need a solid conductor or would a mesh do? 􀂄 Demo D32 (Faraday’s cage in Van der Graaf) 􀂄 Is shielding perfect? E=0 G. Sciolla – MIT 8.022 – Lecture 5 6 􀂄 Consider field lines: 􀂄 Radial around the charge 􀂄 Perpendicular to the surface conductor 􀂄 The point charge +Q induces – charges on the conductor Application of Uniqueness Theorem: Method of images + ---------􀂄 What is the electric potential created by a point charge +Q at a distance y from an infinite conductive plane? 4 G. Sciolla – MIT 8.022 – Lecture 5 7 Method of images 􀂄 Apply the uniqueness theorem 􀂄 It does not matter how you find the potential φ as long as the boundary conditions are satisfied. The solution is unique. 􀂄 In our case: on the conductor surface: φ=0 and always perpendicular 􀂄 Can we find an easier configuration of charges that will create the same field lines above the conductor surface? 􀂄 YES! 􀂄 For this system of point charges we can calculate φ(x,y,z) anywhere 􀂄 This is THE solution (uniqueness) 􀂄 NB: we do not care what happens below the surface of the conductor: that is nor the region under study -+ ---------G. Sciolla – MIT 8.022 – Lecture 5 8 Capacitance +Q + + + + + -Q ------+ 􀂄 What is the ∆φ between the 2? 􀂄 Let’s try to calculate: 􀂄 Caveat: C is proportional to Q only if there is enough Q, uniformly spread… 2 2 1 1 (constant depending on geometry) V E ds Q φ φ ≡ − = − = × ∫ 􀁇 􀁇 i 􀂄 Consider 2 conductors at a certain distance 􀂄 Deposit charge +Q on one and –Q on the other 􀂄 They are conductors 􀃆 each surface is equipotential Q CV ⇒ = 􀂄 Naming the proportionality constant 1/C: 􀂄 Definitions: 􀂄 C = capacitance of the system 􀂄 Capacitor: system of 2 oppositely charged conductors5 G. Sciolla – MIT 8.022 – Lecture 5 9 Units of capacitance 􀂄 Definition of capacitance: 􀂄 Units: 􀂄 SI: Farad (F) = Coulomb/Volt 􀂄 cgs: cm = esu/(esu/cm) 􀂄 Conversion: 1 cm = 1.11 x 10-12 F ~ 1 pF 􀂄 Remember: 􀂄 1 Coulomb is a BIG charge: 1 F is a BIG capacitance 􀂄 Usual C ~ pF-µF V Q CV Q C = ⇒ = G. Sciolla – MIT 8.022 – Lecture 5 10 Simple capacitors: Isolated Sphere 􀂄 Conductive sphere of radius R in (0,0,0) with a charge Q 􀂄 Review questions: 􀂄 Where is the charge located? 􀂄 Hollow sphere? Solid sphere? Why? 􀂄 What is the E everywhere in space? 􀂄 Is this a capacitor? 􀂄 Yes! The second conductor is a virtual one: infinity 􀂄 Calculate the capacitance: 􀂄 Capacitors are everywhere! /R sphere V QR C R Q Q φ φ∞ = − = ⎧ ⇒ = ⎨ = ⎩ R Q6 G. Sciolla – MIT 8.022 – Lecture 5 11 The prototypical capacitor: Parallel plates 􀂄 Physical configuration: 􀂄 2 parallel plates, each of area A, at a distance d 􀂄 NB: if d2<R2 E 1enc E 2enc E 1 2 enc E Gauss's law is the key. on spherical surface with rR . Q =+Q-Q=0 E=0 on spherical surface with R b: E=0 (Gauss) ˆ 2Q a