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# FundamentalsofPhysicsLectureII PHYS187

Drexel

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This 8 page Class Notes was uploaded by Vernice Schuster on Wednesday September 23, 2015. The Class Notes belongs to PHYS187 at Drexel University taught by MichaelCarchidi in Fall. Since its upload, it has received 51 views. For similar materials see /class/212521/phys187-drexel-university in Physics 2 at Drexel University.

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Date Created: 09/23/15

Notes Chapters 25 and 26 Electric Potential and Capacitance 1 The concepts covered in these chapters are 1 Electric potential Electric potential energy Capacitance Capacitors arranged in series and parallel con guration 959 Air and dielectricfilled capacitors 6 Potential energy stored in capacitors 2 Potential and Potential Energy In the previous chapters we learned that charges exert a force on each other according to Coulomb s law This means that such forces among the charges can change a given con guration of charges in the eld view of electrostatic interaction the electric elds of various charges interact to make the charges move ie the electric eld does work on the charge con guration A charge con guration can also be changed by an external agent ie work can be performed by forces external to the charge con guration This change in charge con guration will show up as a change in the potential energy of the system the charge con guration The change in the potential energy between or of two charges is de ned as AW eld AU 1a AWext AU 1b The minus sign in Eq 1a means that when you leave charges alone they will adjust their con guration in such a way as to lower the potential energy of the con guration Eqlb means that if you an external agent perform positive work on the system the change in the potential energy of the system AU will be positive ie you will increase the potential energy of the system Think about this if you let two positive charges alone they will move farther away from each other and thereby reduce their potential energy 0n the other hand the potential energy of two oppositely charged particles is reduced when they move closer together 17 A 4 l7 4 AUab Ua iUb Fdr qudr 2 The potential difference AV Vb Va is defined as b A A Vb39Va jEdr 3 Potential can be de ned as the potential energy per unit charge recall a similar definition of the electric eld E as the force per unit charge qE F A point charge Q creates an electric potential Vr at a distance r from it given by Vr kQr referenced to in nity ie Voo 0 4 The net potential at a given point due to a collection of charges is simply the algebraic sum of individual potentials Vkig 5 21 r Potential energy of two point charges is given by U12 kQ1Q2r 6 Potential energy of a collection or con guration of point charges is the algebraic sum of the potential energy of all the pairs of charges you can form in the con guration For example in Solved Example 1 below U is the potential energy of the 3charge con guration Potential energy of a single charge in a collection or con guration of point charges is the algebraic sum of the potential energy of all the pairs the given charge forms in the con guration For example in Solved Example 1 the potential energy of Q1in part a would be the sum U12 U13 Points to remember 1 Make sure you understand the difference between potential and potential energy and between potential energy of a charge con guration and the potential energy of a charge in a charge configuration 2 A single charge will create a potential at a point in space but you need at least two charges for the potential energy to have any meaning 3 Potential and potential energy are scalars 4 Eq 3 can also be expressed as Ex dVdx and similar relations in the y and zdirections This allows one to calculate the components of E a vector by knowing the potential Vxyz which is a scalar Please see sec 254 of your Text for additional details 3 Capacitance The capacitance C is defined as C QV C is always positive When we talk about the charge on a capacitor we mean the magnitude of the charge on one of its plates The capacitors can be arranged in a series or a parallel arrangement see sec 263 Combination of Capacitors of your Text for diagrams of capacitors in parallel and series combinations In a series arrangement i the net capacitance C is given by JC 1C11C21C3 7 ii The magnitude of the charge on every capacitor is the same iii The potential difference across each capacitor is different and is given by V QCi In a parallel arrangement i The net capacitance C is given by CC1C2C3 8 ii The potential difference Vacross each capacitor is the same iii The charge on every capacitor is different and is given by Q VCi The capacitance of a parallel plate capacitor of plate area A and separation d is given by CA8d 9 The energy stored in a capacitor is given by U IZCVZ 10 By using C QV eq 10 can be cast in terms of other parameters of interest U 12QVor U12Q2C Insertion of a dielectric medium in a capacitor will change its capacitance by afactor of K where K is the dielectric constant of the medium C K KCg 4 Solved Examples 1 Three charges are arranged at the vertices of a rightangle Q1 triangle as shown with Q 4uC 2Q2 and Q3 3uC a Determine the total potential energy of the threecharge system b Determine the work done by an eXtemal force to move Q 2 from its present position to in nity Q3 53 Q2 50cm gt a The initial potential energy 712 712 712 U129X109 12gtltl72 6gtlt1072 8gtlt1072 3X10 5X10 4X10 288J b after Q2 has been removed to in nity the potential energy of the remaining charges is 12gtlt10 12 U 2 9X109 360J f 3x104 the work done by the external force is given by AW AU Uf U 3 607 288 0 72 Note in the problem above we have used AW AU Uf U where AU is the change in the potential energy of the charge configuration We could have also done the problem by looking at it from the point of view of Q2 in a manner of speaking charges don t really have a point of view In this case AU would be the change in the potential energy of Q2 AU UQg UQ2 07 Up U23 9109 12 2010 0 721 2 Consider three charges Q1 Q2 and Q3 placed as shown in the diagram below disregard Q4 for the moment a Calculate the electric potential energy of the three charge configuration b A fourth particle of mass m 2010 13kg and charge Q4 400nC is placed on the X axis 30 cm from Q2 and released from rest Find the speed of Q4 when it has moved very far away from the three charge Q1 Q2 and Q3configuration a The total energy of the three charges is U U12 Uz3 U13 U13 9109400103918210392 45 10395 J Why is U12 U23 0 b U KE U KEf conservation of energy from PHYS101 U KEf U1 is the initial potential energy of Q4 Ui U14 U34 U24 U24 9109 4001039183quot10392 120 10395 J 122103913v2 why is U14 U34 0 v 12 2104ms 3 46104ms c What is the charge on C4 d What is the charge on C1 e What is the total electric energy stored in the fourcapacitor system C1 C2 a I I d b c3 2 3 24 I 6W 4uF 241F d a b b o I C4 a Cad ClCz 01 C2 C3 60uF the resultant of C1 and C2 connected in series is connected in parallel with C3 b Vadde CdbCad 4060 or de 15Vad 1 Also note that Vad de 250V 2 From eqs 1 and 2 Vad 15Vad 250V or Vad 100V and de 150V c Q4 04 de 40pF 150V 600nC d Q1 01 Vadz 40Mquot 50V 200pc Note VadZ is the potential across C1 e U 05C V2 0524nF 2502 7510394J net ab Concept Questions 1 A spherical balloon contains a positively charged object at its center i As the balloon is in ated to a greater volume while the charged object remains at the center does the electric potential at the surface of the balloon a increase b decrease or c remain the same ii Does the electric ux through the surface of the balloon a increase b decrease or c remain the same Answer i b The electric potential is inversely proportional to the radius ii c Because the same number of eld lines passes through a closed surface of any shape or size the electric ux through the surface remains constant Vh1 r 2 i In a certain region of space the electric potential is zero everywhere along the x axis From this information we can conclude that the x component of the electric field in this region is a zero b in the x direction or c in the 7x direction ii In a certain region of space the electric field is zero From this information we can conclude that the electric potential in this region is a zero b constant c positive or 1 negative Answer i a If the potential is constant zero in this case its derivative along this direction is zero ii b If the electric field is zero there is no change in the electric potential and it must be constant This constant value could be zero but it does not have to be zero 3 A capacitor stores charge Q at a potential difference V Ifthe voltage applied by a battery to the capacitor is doubled to 2 V a the capacitance falls to half its initial value and the charge remains the same b the capacitance and the charge both fall to half their initial values c the capacitance and the charge both double or d the capacitance remains the same and the charge doubles Answer d The capacitance is a property of the physical system and does not vary with applied voltage According CQ V if the voltage is doubled the charge is doubled 4 Two capacitors are identical They can be connected in series or in parallel i If you want the smallest equivalent capacitance for the combination a do you connect them in series b do you connect them in parallel or c do the combinations have the same capacitance ii Each capacitor is charged to a voltage of 10 V If you want the largest combined potential difference across the combination a do you connect them in series b do you connect them in parallel or e do the combinations have the same potential difference Answer i a When connecting capacitors in series the inverse of the capacitances add resulting in a smaller overall equivalent capacitance ii a When capacitors are connected in series the voltages add for a total of 20 V in this case If they are combined in parallel the voltage across the combination is still 10 V 5 You have three capacitors and a battery In which of the following combinations of the three capacitors will the maximum possible energy be stored when the combination is attached to the battery a When in series the maximum amount is stored b When parallel the maximum amount is stored c Both combinations will store the same amount of energy Answer b For a given voltage the energy stored in a capacitor is proportional to C U CV22 Therefore you want to maximize the equivalent capacitance You do so by connecting the three capacitors in parallel so that the capacitances add 6 When the potential difference between the plates of a capacitor is doubled the magnitude of the electric energy stored in the capacitor a is doubled b is halved 0 remains the same d quadruple 7 Three charges are placed on the three corners of an equilateral triangle as shown The total potential energy of these charges is a b715kQ2r c zero d15kQ2r

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