Capacitance And Dielectrics
Capacitance and DielectricsPhysics 231Lecture 4-1Fall 2008
CapacitorsDevice for storing electrical energy which can then bereleased in a controlled mannerConsists of two conductors, carrying charges of q and –q,that are separated, usually by a nonconducting material - aninsulatorSymbol in circuits isIt takes work, which is then stored as potential energy in theelectric field that is set up between the two plates, to placecharges on the conducting plates of the capacitorSince there is an electric field between the plates there is also apotential difference between the platesPhysics 231Lecture 4-2Fall 2008
CapacitorsWe usually talk aboutcapacitors in terms ofparallel conductingplatesThey in fact can beany two conductingobjectsPhysics 231Lecture 4-3Fall 2008
CapacitanceThe capacitance is defined to be the ratio of theamount of charge that is on the capacitor to thepotential difference between the plates at this pointQC V abUnits arePhysics 2311Coulomb1 farad 1VoltLecture 4-4Fall 2008
Calculating the CapacitanceWe start with the simplest form – two parallel conductingplates separated by vacuumLet the conducting plates have area A andbe separated by a distance dThe magnitude of the electric fieldbetween the two plates is given byWe treat the field as being uniformallowing us to writePhysics 231Lecture 4-5QσE ε0 ε0AVabQd Ed ε0AFall 2008
Calculating the CapacitancePutting this all together, we have for thecapacitanceQA ε0C VabdThe capacitance is only dependentupon the geometry of the capacitorPhysics 231Lecture 4-6Fall 2008
1 farad CapacitorGiven a 1 farad parallel plate capacitor having aplate separation of 1mm. What is the area of theplates?AWe start with C ε 0dAnd rearrange tosolve for A, givingA Cdε0(1.0F )(1.0 10 3 m ) 8.85 10 12 F / m 1.1 108 m 2This corresponds to a square about 10km on a side!Physics 231Lecture 4-7Fall 2008
Series or Parallel CapacitorsSometimes in order to obtain needed valuesof capacitance, capacitors are combinedin eitherSeriesorParallelPhysics 231Lecture 4-8Fall 2008
Capacitors in SeriesCapacitors are often combined in series and the questionthen becomes what is the equivalent capacitance?Givenwhat isWe start by putting a voltage,Vab, across the capacitorsPhysics 231Lecture 4-9Fall 2008
Capacitors in SeriesCapacitors become charged because of VabIf upper plate of C1 gets a charge of Q,Then the lower plate of C1 gets acharge of -QWhat happens with C2?Since there is no source of charge at point c, and wehave effectively put a charge of –Q on the lower plateof C1, the upper plate of C2 gets a charge of QCharge ConservationThis then means that lower plate of C2 has a charge of -QPhysics 231Lecture 4-10Fall 2008
Capacitors in SeriesWe also have to have that the potential across C1 plus thepotential across C2 should equal the potential drop acrossthe two capacitorsVab Vac Vcb V1 V2We haveThenQQV1 and V2 C1C2VabQ Q C1 C 2Dividing through by Q, we havePhysics 231Lecture 4-11Vab 11 Q C1 C 2Fall 2008
Capacitors in SeriesThe equivalent capacitor will also have thesame voltage across itVab 11 Q C1 C 2The left hand side is the inverse of 1 V the definition of capacitanceC QSo we then have for the equivalent capacitance111 C eq C1 C 2If there are more than two capacitors in series,the resultant capacitance is given by11 C eqi CiPhysics 231Lecture 4-12Fall 2008
Capacitors in ParallelCapacitors can also be connected in parallelGivenwhat isAgain we start by puttinga voltage across a and bPhysics 231Lecture 4-13Fall 2008
Capacitors in ParallelThe upper plates of both capacitorsare at the same potentialLikewise for the bottom platesWe have thatNowV1 V2 VabQ1Q2and V2 V1 C1C2orQ1 C1V and Q2 C 2VPhysics 231Lecture 4-14Fall 2008
Capacitors in ParallelThe equivalent capacitor will have the same voltage acrossit, as do the capacitors in parallelBut what about the charge on the equivalent capacitor?The equivalent capacitor will have the same total chargeQ Q1 Q2Using this we then haveQ Q1 Q2C eqV C1V C 2VorC eq C1 C 2Physics 231Lecture 4-15Fall 2008
Capacitors in ParallelThe equivalent capacitance is just the sum of thetwo capacitorsIf we have more than two, the resultant capacitance is justthe sum of the individual capacitancesC eq C iiPhysics 231Lecture 4-16Fall 2008
Example 1aC3C1C2C abbWhere do we start?Recognize that C1 and C2 are parallel with each other andcombine these to get C12This C12 is then in series with with C3The resultant capacitance is then given by1 11 C C3 C1 C2Physics 231 Lecture 4-17C 3 (C1 C 2 )C C1 C 2 C 3Fall 2008
Example 2CCCCCConfiguration AConfiguration BConfiguration CThree configurations are constructed using identical capacitorsWhich of these configurations has the lowest overall capacitance?a) Configuration Ab) Configuration Bc) Configuration CThe net capacitance for A is just CIn B, the capsare in series andthe resultant isgiven by11 1 2C C net C net C C C2In C, the caps are in parallel andthe resultant is given byPhysics 231Lecture 4-18C net C C 2 CFall 2008
Example 3A circuit consists of three unequal capacitors C1, C2, and C3which are connected to a battery of emf Ε. The capacitorsobtain charges Q1 Q2, Q3, and have voltages across theirplates V1, V2, and V3. Ceq is the equivalent capacitance of thecircuit.Check all of the following thatapply:a) Q1 Q2b) Q2 Q3c) V2 V3d) E V1e) V1 V2f) Ceq C1A detailed worksheet is available detailing the answersPhysics 231Lecture 4-19Fall 2008
Example 4oWhat is the equivalentcapacitance, Ceq, of thecombination shown?(a) Ceq (3/2)CCC11 1 C1 C CPhysics 231CeqCCCo(b) Ceq (2/3)CC CC1 2Lecture 4-20(c) Ceq 3CC1CC eqC 3 C C2 2Fall 2008
Energy Stored in a CapacitorElectrical Potential energy is stored in a capacitorThe energy comes from the work that is done in chargingthe capacitorLet q and v be the intermediate charge and potential on thecapacitorThe incremental work done in bringing an incrementalcharge, dq, to the capacitor is then given byq dqdW v dq CPhysics 231Lecture 4-21Fall 2008
Energy Stored in a CapacitorThe total work done is just the integral of thisequation from 0 to Q1W CQ 0Q2q dq 2CUsing the relationship between capacitance, voltage andcharge we also obtainQ2 112U C V QV2C 22where U is the stored potential energyPhysics 231Lecture 4-22Fall 2008
Example 5Suppose the capacitor shown here is chargedto Q and then the battery is disconnectedNow suppose you pull the plates furtherapart so that the final separation is d1Which of the quantities Q, C, V, U, E change?A d-----Q: Charge on the capacitor does not changeC: Capacitance DecreasesV: Voltage IncreasesU: Potential Energy IncreasesE: Electric Field does not changeHow do these quantities change?Answers:Physics 231C1 dCd1V1 Lecture 4-23d1VdU1 d1UdFall 2008
Example 6Suppose the battery (V) is keptattached to the capacitorAgain pull the plates apart from d to d1Now which quantities, if any, change?Q: Charge DecreasesC: Capacitance DecreasesV: Voltage on capacitor does not changeU: Potential Energy DecreasesE: Electric Field DecreasesHow much do these quantities change?Answers:Physics 231dQ1 Qd1dC1 Cd1Lecture 4-24dU1 Ud1dE1 Ed1Fall 2008
Electric Field Energy DensityThe potential energy that is stored in the capacitor can bethought of as being stored in the electric field that is in theregion between the two plates of the capacitorThe quantity that is of interest is in fact the energy densityEnergy Density u 1CV 22Adwhere A and d are the area of the capacitor plates and theirseparation, respectivelyPhysics 231Lecture 4-25Fall 2008
Electric Field Energy DensityAUsing C ε 0and V E d we then haved12u ε0 E2Even though we used the relationship for a parallel capacitor,this result holds for all capacitors regardless of configurationThis represents the energy density of the electric field ingeneralPhysics 231Lecture 4-26Fall 2008
DielectricsMost capacitors have a nonconducting material betweentheir platesThis nonconducting material, a dielectric, accomplishesthree things1) Solves mechanical problem of keeping the platesseparated2) Increases the maximum potential difference allowedbetween the plates3) Increases the capacitance of a given capacitor overwhat it would be without the dielectricPhysics 231Lecture 4-27Fall 2008
DielectricsSuppose we have a capacitor of value C0 that is charged toa potential difference of V0 and then removed from thecharging sourceWe would then find that it has a charge ofQ C 0V0We now insert the dielectric material into the capacitorWe find that the potential difference decreases by a factor KV0V KOr equivalently the capacitance has increased by a factor of KC K C0This constant K is known as the dielectric constant and isdependent upon the material used and is a numbergreater than 1Physics 231Lecture 4-28Fall 2008
PolarizationQ Without the dielectric inthe capacitor, we haveV0E0---------------The electric field points undiminished from the positiveto the negative plateQ -E -- V κ With the dielectricin place we have---------------The electric field between the plates of the capacitor isreduced because some of the material within the dielectricrearranges so that their negative charges are orientedtowards the positive platePhysics 231Lecture 4-29Fall 2008
PolarizationThese rearranged charges set up aninternal electric field that opposesthe electric field due to the chargeson the platesThe net electric field is given byE0E KPhysics 231Lecture 4-30Fall 2008
RedefinitionsWe now redefine several quantities using the dielectricconstantWe define the permittivity of the dielectric as beingε K ε0AAC KC 0 Kε 0 εddwith the last two relationships holding for a parallelplate capacitorCapacitance:Energy DensityPhysics 23111 22u Kε 0 E εE22Lecture 4-31Fall 2008
Example 7Two identical parallel plate capacitors are given the same chargeQ, after which they are disconnected from the battery. After C2has been charged and disconnected it is filled with a dielectric.Compare the voltages of the two capacitors.a) V1 V2b) V1 V2c) V1 V2We have that Q1 Q2 and that C2 KC1We also have that C Q/V or V Q/CThenPhysics 231V1 Q1C1andV2 Q2Q1 1 V1C 2 KC1 KLecture 4-32Fall 2008
Symbol in circuits is It takes work, which is then stored as potential energy in the electric field that is set up between the two plates, to place charges on the conducting plates of the capacitor Since there is an electric field between the plate
Stacked Dielectrics Consider a parallel-plate capacitor with areaA of each plate and spacing d. Capacitance without dielectric: C0 ǫ0A d. Dielectrics stacked in parallel: C C1 C2 with C1 κ1ǫ0 A/2 d, C2 κ2ǫ0 A/2 d. C 1 2 (κ1 κ2)C0. Dielectrics stacked
RF Capacitance level controls are based on an electronic device called a capacitor. The capacitor is a device that stores energy. This energy is not stored in the probe; rather, the RF Capacitance level control is merely measuring how much energy can be stored. The amount of capacitance the RF Capacitance
P2. Solutions to Problems: Dielectrics, Capacitance, Electric Energy 51 O z a b h h E 1 E 2 E r ps1 r ps2 Figure P2.2 Evaluation of the electric field intensity vector at the center of a
Dielectrics and Polarization 11/24/2017 6 Dielectrics may be subdivided into two groups : Non-Polar: dielectrics that do not possess permanent electric dipole moment. Electric dipole moments can be induced by placing the materials
i. Active dielectrics When a dielectric material is kept in an external electric field, if it actively accepts the electricity, then it is known as active dielectric material. Thus, active dielectrics are the dielectrics, which can easily adapt themselves to store the electric
dielectrics and potentially further decreasing the dielectrics of the oil-contaminated sea ice. To better understand the impact of oil on sea ice dielectrics, an artificial oil-in-ice mesocosm experiment was designed, in which the physical migration of light-medi
former oil. The oil switch has many desirable qualities qualifying it for consideration as a repetitive switch medium. Liquid dielectrics when pulse charged have breakdown fields substantially higher than gaseous dielectrics[!]. Liquid dielectrics also have an are.a a d time depend
Electrodynamics of the Dielectrics . By F. F. Mende . Abstract- In the scientific literature is in sufficient detail opened the role of the kinetic inductance of charges in the conductors and the plasmo-like media, but it is not opened the role of this parameter in the electrodynamics of dielectrics. This parameter in the electrodynamics of
