PHY2054: Chapter 16 Capacitance1CapacitancePHY2054: Chapter 16 Capacitance2Purpose of CapacitorsÎStorage of charge: Q = CV Used in DC and AC circuitsÎStorage of energy Can provide energy to circuitsÎUsed in DC and AC circuits Timing in DC circuits Resonance in AC circuits (Later in course)22122QUCVC==PHY2054: Chapter 16 Capacitance3Capacitors in ParallelÎV1= V2= V3(same potential top and bottom)ÎTotal charge: qtot= q1+ q2+ q3ÎCeqV= C1V + C2V + C3V¾ Basic law for combiningcapacitors in parallel¾ Works for N capacitorseq 1 2 3CCCC=++VVPHY2054: Chapter 16 Capacitance4Capacitors in SeriesÎq1= q2= q3(same current charges all capacitors)ÎTotal potential: V = V1+ V2+ V3Îq/Ceq= q/C1+ q/C2+ q/C3eq 1 2 31111CCCC=++¾ Basic law for combiningcapacitors in series¾ Works for N capacitorsVVPHY2054: Chapter 16 Capacitance5ConcepTestÎTwo identical parallel plate capacitors are shown in an end-view in Figure A. Each has a capacitance of C.If the two are joined together at the edges as in Figure B, forming a single capacitor, what is the final capacitance? (a) C/2 (b) C (c) 2C (d) 0 (e) Need more informationA BArea is doubledPHY2054: Chapter 16 Capacitance6ConcepTestÎEach capacitor is the same in the three configurations. Which configuration has the lowest equivalent capacitance? (1) A (2) B (3) C (4) They all have identical capacitanceCCCCCAB CC/2 (series)PHY2054: Chapter 16 Capacitance7Capacitors in CircuitsÎFind total capacitance Ceqbetween (a) and (b) Use multi-step processÎC23= 2 + 3 in series 1/C23= 1/C + 1/C = 2/C C23= C/2ÎC1+ C23in parallel Ceq= C + C/2 = 3C/2ooCCCCeq123(a)(b)PHY2054: Chapter 16 Capacitance8Another ExampleÎAssume all capacitors = 10 μF. Find total capacitance C3and C4in parallel: C34= 10 + 10 = 20 μF C1, C34, C2in seriesÎHow much charge provided by battery to fully charge capacitors? Assume V = 20. Q = Ceqx V = 4 x 20 = 80 μCeqeq1 111 510 20 10 204.0μFCC=++==PHY2054: Chapter 16 Capacitance9Another ExampleÎAssume all capacitors = 10 μF. Find total capacitance C234= 30 μF (parallel) C56= 20 μF (parallel) C1, C234, C56(series)ÎHow much charge provided by battery to fully charge capacitors? Assume V = 20. Q = Ceqx V = 5.45 x 20 = 109 μCeqeq11111110 30 20 605.45μFCC=++ ==PHY2054: Chapter 16 Capacitance10Find Charges on Series CapacitorsÎLet V = 10, C1= 6μF, C2= 12μF Find charges, voltages on C1, C2ÎCombine series capacitances This is what battery “sees”!ÎFind qeq, then use qeq= q1= q2qeq= CeqV = 4 x 10 = 40 μCÎFind V1, V2V1= q1/ C1= 40 / 6 = 6.67 V V2= q2/ C2= 40 / 12 = 3.33 V V1+ V2= 10, as expected12eq12724μF18CCCCC===+PHY2054: Chapter 16 Capacitance11Example: Find qiand Vion All CapacitorsÎC1is charged in position A, then S is thrown to B position Initial voltage across C1:V0= 12 Initial charge on C1:q10= 12 x 4 = 48μCÎAfter switch is thrown to B: Charge flows from C1to C2and C3 V1= V23(parallel branches)Îq2and q3in series: q2= q3= q23(C23= 2μF) Charge conservation: q10= q1+ q23 48 = C1V1+ C23V1(V1= V23) Find V1: V1= 48 / (C1+ C23) = 8 V Find q1: q1= C1V1= 32μC q23= 48 – 32 = q2= q3= 16μC V2= q2/ C2= 2.67 V V3= q3/ C3= 5.33 V4μF6μF3μF12VABPHY2054: Chapter 16 Capacitance12Another ExampleÎEach capacitor has C = 10μF. Find the total capacitanceÎDo it in stages 2 & 3 ⇒ C23= 5 μF Add 4 ⇒ C234= 15 μF Add 5 ⇒ C2345= 6 μF Add 1 ⇒ C12345= 16 μFÎCharge supplied by battery (20V) qtot= C12345x V = 16 x 20 = 320 μC12345PHY2054: Chapter 16 Capacitance1312345Find Charges, Voltages on All CapacitorsÎEach capacitor has capacitance 10μF. V = 20 volts q1= C1V = 10 x 20 = 200μCÎC2345= 6μF, q2345= 6 x 20 = 120 μC q2345= q234= q5= 120μC (series) V5= q5/ C5= 120 / 10 = 12ÎFind q4, V4= V23V234= V4= 20 – 12 = 8 q4= C4V4= 10 x 8 = 80μCÎFind q2, q3, V2, V3(C23= 5μF) q2= q3= q23= C23x V23= 5 x 8 = 40μC V2= q2/ C2= 40 / 10 = 4 V3= q3/ C3= 40 / 10 = 4Check: V2+ V3= 8 (= V23)PHY2054: Chapter 16 Capacitance14Capacitor MonsterAll voltages = 4V, all capacitors = 2μF. What is the charge on C?Can you find the charge on all capacitors?PHY2054: Chapter 16 Capacitance15Capacitor MonsterAll voltages = 4V, all capacitors = 2μF. What is the charge on C?q = CV = 2 x 4 = 8μC00044444440404044PHY2054: Chapter 16 Capacitance16Energy in a CapacitorÎCapacitors have energy associated with them Grab a charged capacitor with two hands and find out!ÎCalculation of stored energy Proof requires simple calculus derivation Energy = work moving charge from – to + surfaceÎCapacitors store and release energy as they acquire and release charge This energy is available to drive circuits22122QUCVC==PHY2054: Chapter 16 Capacitance17Example of Capacitor EnergyÎC = 5 μF, V = 200ÎChange V to 20000 (as in demo of large HV capacitor)()26211225 10 200 0.1JUCV−==×××=()26211225 10 20000 1000JUCV−==××× =PHY2054: Chapter 16 Capacitance18Where is the Energy Stored?ÎAnswer: Energy is stored in the electric field itself!!ÎExample: Find energy density of two plate capacitor E field is constantÎEnergy density depends only on E field! A general result, independent of geometry Can be shown more generally by Maxwell’s equations()()2220102/22Ad EdUCVuEAd Ad Adεε== = =2102uEε=PHY2054: Chapter 16 Capacitance19Dielectric Materials and CapacitorsÎInsulating material that can be polarized in E fieldÎInduced charges at dielectric surface partially cancel E field E → E / κκ> 1 is “dielectric constant” V → V / κ (since V = Ed) C →κC (since C = Q / V)ÎBut “good” dielectric requires more than high κ value Good insulator (no charge leakage) High breakdown voltage (no arcing at high voltage) Low cost (affordable)Dielectricmaterial++++++++++++++++++++++++++++++++++++++++++++++++++--------------------------------------------------------------------------------------κInducedchargesPHY2054: Chapter 16 Capacitance20Dielectric Mechanism is Due to PolarizationE = 0, Dipoles randomly aligned¾ E applied, partially aligns dipoles¾ Aligned dipoles induce surface charges¾ Surface charges partially cancel E fieldhttp://hyperphysics.phy-astr.gsu.edu/hbase/electric/dielec.htmlPHY2054: Chapter 16 Capacitance21Dielectric Constants of Some Materials3.8Silica glass2.7 – 3.0Beeswax3.7Paper100 – 1250Barium