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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