Electricity and Magnetism--Current Electricity

Topics: 􀂄 􀂄 Conductivity and resistivity 􀂄 8.022 (E&M) – Lecture 7 Electrical currents Ohm’s law in microscopic and macroscopic form 2 Electric current I 􀂄 􀂄 i􀂄 􀂄 Units: 􀂄 cgs: esu/s 􀂄 C/s=ampere (A) 􀂄 9 dQI dt = G. Sciolla – MIT 8.022 – Lecture 7 Consider a region in which there is a flow of charges: E.g. cylndrical conductor We define a current: the charge/unit time flowing through a certain surface SI: Conversion: 1 A = 2.998 x 10esu/s 13 Current density J 􀂄 􀂄 Velocity of each charge: u 􀂄 ∆Q /∆t 􀂄 where ∆􀃆 􀂄 cosprismqnV qnAI qnt t t t θ∆ ∆ ∆ = = = = = = ∆ ∆ ∆ ∆ 􀁇 􀁇􀁇􀁇i i A 􀁇 ut∆ θ J qnu uρ≡ ≡ 􀁇 􀁇 􀁇 lume charge densiG. Sciolla – MIT 8.022 – Lecture 7 Number density: n = #charges /unit volume Current flowing through area A: I = Q= q x number of charges in the prism Where we defined the current density J as: Q q N u t uA JA ΝΒ: ρ=voty 4 More realistic case… 􀂄 􀂄 ii􀂄 u 􀂄 􀂄 Multiple charge carriers: 􀂄 liind of i􀂄 k iik 􀂄 􀂄 i􀃆 average velocity 􀂄 J: kkk kk k k J uρ≡ ≡∑ ∑􀁇 􀁇 􀁇 1 ()k ik u uN = ∑􀁇 􀁇 kk k k k k k J uρ≡ ≡∑ ∑􀁇 􀁇 􀁇 S I= ∫ 􀁇􀁇 i G. Sciolla – MIT 8.022 – Lecture 7 We made a number of unrealistic assumptions: only 1 knd of charge carriers: we could have several, e.g.: + and – ons assumed to be the same for all particles: unrealistic! regular surface with J constant on it E.g.: souton with different kons NB: + ion with velocity uis equivalent to – on with velocty -uVelocity: Not all charges have the same velocty Arbitrary surface S, arbitrary qnu k i qn u JdA 25 Non standard currents 􀂄 􀂄 􀂄 Other kinds of currents 􀂄 (Demo F5) + + + + ----Na+ Cl-G. Sciolla – MIT 8.022 – Lecture 7 We usually think of currents as electrons moving inside a conductor This is only one of the many examples! Ions in solution such as Salt (NaCl) in water 6 The continuity equation 􀂄 􀂄 􀂄 Some charge exits 􀂄 􀂄 􀂄 􀂄 S J J J J J JS Q t ∂=− ∂∫ 􀁇􀁇 i􀁶 00S V V V J tQt t J t ρ ρ ρ ⎧ = ∇ ⎪ ∂⎪ ⎛ ⎞ ⇒ ∇ + = ⇒⎨ ⎜ ⎟∂ ∂ ∂ ∂∇ + ⎝ ⎠⎪ = − = ∂ ∂⎪⎩ ∂− ∫ ∫ ∫ ∫ 􀁇􀁇 􀁇 i i 􀁇􀁇 i i 􀁶 􀁶 G. Sciolla – MIT 8.022 – Lecture 7 A current I flows through the closed surface S: Some charge enters What happens to the charge after it enters? Piles up inside Leaves the surface NB: -because dA points outside the surface Apply Gauss’s theorem and obtain continuity equation: inside JdA inside JdA JdV dV dV 37 Thoughts on continuity equation 􀂄 􀂄 􀂄 􀂄 For steady currents: 􀂄 ρ 􀃆 S J J J J J J 0J t ρ∂∇+ = ∂ 􀁇 i 0J∇ = 􀁇 i G. Sciolla – MIT 8.022 – Lecture 7 Continuity equation: What does it teach us? Conservation of electric charges in presence of currents no accumulation of charges inside the surface: d/dt=0 8 Microscopic Ohm’s law 􀂄 Electric fields cause charges to move 􀂄 􀂄 􀂄 􀂄 conductivity σ J Eσ= 􀁇 􀁇 e G. Sciolla – MIT 8.022 – Lecture 7 Experimentally, it was observed by Ohm that Microscopic version of Ohm’s law: It reflects the proportionality between E and J in each point Proportionality constant: More stuff here pleas 49 􀂄 in E //L 􀂄 􀂄 σMacroscopic Ohm’s lawRI V LV IJ E A AL Rσ σ σ =⇒ ≡= ⇒ = A L J E G. Sciolla – MIT 8.022 – Lecture 7 Current is flowing in a uniform material of length L uniform electric field Potential difference between two ends: V=EL Ohm’s law J=E holds in every point: where Resistance R 􀂄 Proportionality constant between V and R in Ohm’s law R ≡ L Aσ ≡ L A ρ 􀂄 Units: [V]=[R][I] 􀂄 SI: Ohm (Ω) = V/A 􀂄 cgs: s/cm 􀂄 Dependence on the geometry: 􀂄 Inversely proportional to A and proportional to L 􀂄 Dependence on the property of the material: 􀂄 Inversely proportional to conductivity G. Sciolla – MIT 8.022 – Lecture 7 10 5 􀂄 Resistivity ρ = 1/σ 􀂄 􀂄 Units: in SI: Ω 􀂄 Depends on chemistry of material, ,… 􀂄 Demos F1 and F4 Resistivity G. Sciolla – MIT 8.022 – Lecture 7 11 Describes how fast electrons can travel in the material m; in cgs: s temperature􀂄 􀂄 Demos F1 and F4 􀂄 Why? 􀂄 Room temperature: 􀂄 ρ 􀃆 when T i􀃆 􀃆 ρ increases 􀂄 Very low temperature: 􀂄 material 􀃆 􀂄 Resistivity vs. Temperature ρ T G. Sciolla – MIT 8.022 – Lecture 7 12 Does resistivity depend on T? depends upon collisional processes ncreases more collisions Mean free path dominated by impurities or defects in the ~ constant with temperature. With sufficient purity, some metals become superconductors 6􀂄 σ E 􀂄 Spherical symmetry 􀃆 􀂄 φ φ 􀃆 􀂄 φ ): 􀂄 Application: Resistance of a spherical shell 􀂄 ρ 􀂄 Difference in potential V 􀃆 current 􀂄 φ φ =0 Q: what is the resistance R? b a V ρ () B r A r φ = + 1() ab a r Vφ ⎛ ⎞ = −⎜ ⎟− −⎝ ⎠ 2 2 1 1ˆ() ab abEr V r J Vσ= ⇒ = − − 􀁇 4 44 ab V VI V abba I V ba baR abπσ πσ πσ = = = ⇒ = = = − − −∫ 􀁇 􀁇􀁇 􀁇 i i G. Sciolla – MIT 8.022 – Lecture 7 13 Microscopic Ohm will hold: J=spherical potential: Boundary conditions: (a)=V and (b)=0 E=-grad(2 concentric spheres; material in between has resistivity inner=V; outerV=0 ba r b a ba r b a r Sphere J dA J A What if σ is not constant? 􀂄 ivity σ 1 and σ 2 􀂄 􀂄 􀃆 Eli􀃆 i􀃆 σ q l11 2 2I AEσ σ= = σ 1 σ 2 I 2 1 2 1( ) 4 4 4q E E E I A ρ ρσ π π π − − = = = G. Sciolla – MIT 8.022 – Lecture 7 14 Cylindrical wire made of 2 conductors with conductWhat is the consequence? Current flowing must be the same in the whole cylinder ectric fields are dfferent in the 2 regions E dscontinuous surface layer at the boundary When conductivity changes there is the possibility that some charge accumulates somewhere. This is necessary to maintain steady fow. A E surface 7Thoughts on Ohm’s law 􀂄 􀂄 In plain Engli􀂄 A constant el􀂄 􀂄 􀂄 ll􀂄 i and are scattered 􀃆 the average behavior is a uniform drift J Eσ= 􀁇 􀁇 E v∝ 􀁇 􀁇 F ma E a= ⇒ ∝ 􀁇 􀁇􀁇 􀁇 E G. Sciolla – MIT 8.022 – Lecture 7 15 Ohm’s law in microscopic formulation: sh: ectric field creates a steady current: Does this make sense? Charges are moving in an effectively viscous medium As sky diver in free fa: first accelerate, then reach constant v Why? Charges are accelerated by E but then bump into nucleMotion of electrons in conductor E 􀂄 􀂄 i0: 􀂄 Impull􀂄 The average momentum is: 􀂄 􀃆 􀂄 ii􀂄 ial 0p = 􀁇 􀁇 Ep = 􀁇􀁇 1 1 1 1 1 1( ) N N N i i i i i i i p m u qE tN N N= = = = = + = +∑ ∑ ∑􀁇 􀁇􀁇 􀁇 1 0 N i i u = →∑ 􀁇 1 1 N i i t N τ = ≡ ∑ 1 N i i qEmu tN τ = = ≡∑ 􀁇 􀁇 G. Sciolla – MIT 8.022 – Lecture 7 16 N electrons are moving in a material immersed in Two components contribute to the momentum: Random collision velocty use due to eectric field: For large N: Where is the average tme between 2 collsions Property of the materRandom mu qEt m u mu qEt qE 8Conductivity 􀂄 􀂄 For multiple carriers: 2 = u qE nqE m mmu τ τσ σ τ ⎧ =⎪ ⇒ = ⇒ =⎨ =⎪⎩ 􀁇 􀁇􀁇 􀁇 􀁇 􀁇􀁇 2 1 N kk k i k nq m τσ = =∑ G. Sciolla – MIT 8.022 – Lecture 7 17 From this derivation we can read off the conductivity Jnq Jnq qE 9