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Physics 11bLecture #6CapacitanceS&J Chapter 26What We Did Last Time Electric potential due to continuous charge distributions Use Electric field/potential due tospherical charge distribution Looks like a point charge from outside Zero inside Discussed conductors Electric shielding (Faraday cage) Millikan’s oil-drop experiment01()4dqVrπε=∫rxa04Qaπε04Qxπεxa204Qaπε204Qxπε0()Vx()xExToday’s Goals Define capacitance Two conductors with electric charge Æ What is the potential difference between them? Study parallel-plate capacitor Will also do a cylindrical one Combination of capacitors Æ Rules for “additions” Depend on the configuration: parallel and serial Discuss energy stored in a capacitor And where the energy isPotential Between Conductors We’ve learned: Electric potential V of a contiguous conductor is constant Electric field E exists only outside conductors Surface charge σof a conductor is proportional to E just outside the surface Consider two charged conductors What’s the ∆V = V1– V2between them?++++++−−−−−−q = +qV = V1q = −qV = V2Capacitance ∆V must be proportional to q Define the capacitance Cbetween the two conductors It’s the amount of electric charge required to produce unit difference in electric potential Unit: Too big for practical use Î Use µF = 10-6F, pF = 10-12F++++++−−−−−−+q, V1−q, V2Vq∆∝qCV=∆qCV≡∆orCoulombFaradVolt=Parallel Plate Capacitor Two metallic plates placed close to each other Most common form of capacitors Area A, spacing d E between the plate is uniform Not at the edge, but that’s a smalleffect if d is small compared withthe size of the plates From Lecture #3, we know the relationbetween E and the charge density+Q −Qd++++++++−−−−−−−−00QEAσεε==0QdVEdAε∆= =0AQCVdε==∆ERemember this one?From Lecture #3 Charge can only be on the surface Let’s call the charge density σ= Q/area NB: σmay not be constant for the whole surface Apply Gauss on a cylinder stickingout of the metal surface Φtotal= Φtop+ Φside+ ΦbottomAE= 0total topAEΦ=Φ=inqAσ=0Eσε=Rule #3: E field just outside a conductor in electrostatic equilibrium is perpendicular to the surface and the magnitude is σ/ε0Parallel Plate Capacitor Capacitance of a parallel-plate capacitor Does it make sense? Capacitor with larger A should hold morecharge, because the E field goes with thearea density of the charge Capacitor with larger d should have morepotential ∆V across it, because it’s E timesthe distance d To make a “large” capacitor, you need large plates held together very closely+Q −Qd++++++++−−−−−−−−0AQCVdε==∆E Proportional to the area Inversely prop. to the gapCylindrical Capacitor Roll up a parallel plate capacitor into cylinders To save space… Use Gauss’s Law Imagine a cylindricalsurface of radius r Top and bottomparallel to E Æ Ignore This must equal to Q/ε0abrtotal side2ErπΦ=Φ=× Aℓ+Q−Q02QErπε=A02QErπε=AWeaker at larger rCylindrical Capacitor Integrate E to get ∆V Note E is parallel to rabr[]000ln22ln2bbbaaaQdr QVEdr rrQbaπε πεπε∆= = =⎛⎞=⎜⎟⎝⎠∫∫AAA02ln( )QCVbaπε==∆ADoes thismake sense?Cylindrical Capacitor We’ve found What if the gap is very narrowi.e., d = b – a is small Taylor expansion tells usabr02ln( )QCVbaπε==∆Aln ln ln 1bad daa a+⎛⎞ ⎛ ⎞ ⎛ ⎞==+⎜⎟ ⎜ ⎟ ⎜ ⎟⎝⎠ ⎝ ⎠ ⎝ ⎠1(ln )ln 1 ln(1)xddxddadxaa=⎛⎞+≈ + ×=⎜⎟⎝⎠002 aACddπεε≈=A2 Area of the inner cylinderAaπ==ASame as parallel-plateCapacitor Arithmetic Electrical circuits get tedious to draw Let’s consider combinations of capacitorsshorthandC∆V+−Parallel SeriesParallel Combination Connect two capacitors C1and C2in parallel Potential ∆V across C1and C2are the same C1and C2hold charges Combined, they looks like a capacitorthat holds Parallel Capacitor Rule:∆V+−C1C211QCV=∆22QCV=∆12 12()QQQ C C V=+= + ∆∆V+−Cequiv. 1 2CCC=+Series Combination Connect two capacitors C1and C2in series Charge Q in C1and C2are the same Can you tell why? C1and C2hold charges Total potential ∆V must be sum of∆V1and ∆V2 Series Capacitor Rule:∆V+−C1C211QCV=∆22QCV=∆1212QQVVVCC∆=∆+∆ = +∆V+−Cequiv. 1 2111CCC=++Q−Q+Q−QCapacitor Arithmetic For arbitrary number of capacitors Ex:C1C2C3Cn…C1C2C3Cn…equiv. 1 2 3 nCCCC C=++++"equiv. 1 2 31111 1nCCCC C=++++"2µF3µF5µF5µF5µF2.5µF==Energy in a Capacitor Capacitor C is holding charge q We move small charge dq from thenegative plate to the positive plate Moving charge across potentialrequires work: Question: starting from q = 0, how much work is needed to charge up the capacitor until q = Q? Answer:C+q−q∆VdqqdW dq V dqC=∆=2201()22QqQWdq CVCC===∆∫Capacitor must bestoring this energyCapacitor as a Storage Device Capacitor can hold energy Not a lot for typicalcapacitance and voltage found in electric circuits It can get big with high-voltage circuits Don’t open up your TV set! It may be sufficient for low-current circuits “Battery-less” LED flashlights that light up by shaking  Capacitors can be used for storing “information” Charge up Æ Check the voltage later Computer memory chips (dynamic RAMs) contain arrays of microscopic capacitors21()2UCV=∆Where is the Energy When a capacitor is charged up, there isenergy U in it Exactly where? A capacitor is empty inside All what’s inside is E field Let’s assume the energy is uniformlydistributed between the plates+Q −Qd++++++++−−−−−−−−E21()2UCV=∆2220() ()Energy density 2212UCV CEduAdAd AdEε∆== ==Area A0AQCVdε==∆Empty space with electric field E is filled with energy density 2012EεEnergy in the Field Empty space holding energy may sound strange Consider potential energy due to gravity Potential energy is not in the object, but in its relationship with the gravitational field around it Gravitational field g (like E field) has energy density that is proportional to g2 It’s just another way of looking at forceshmgÍ This object has a potential energy mghBut how does it store the energy?The object itself does not change its properties!Summary Defined capacitance


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HARVARD PHYS 11b - Lecture 6

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