Electric Potential IIElectric Potential IIPhysics 2102Jonathan DowlingPhysics 2102Physics 2102Lecture:Lecture: 09 MON 02 FEB09 MON 02 FEBCh24.6-10PHYS2102 FIRST MIDTERM EXAM!6–7PM THU 05 FEB 2009Dowling’s Sec. 2 in Lockett Hall, Room 6YOU MUST BRING YOUR STUDENT ID!The exam will cover chapters 21 through 24, ascovered in homework sets 1, 2, and 3. The formulasheet for the exam can be found here:http://www.phys.lsu.edu/classes/spring2009/phys2102/formulasheet.pdfTHERE WILL BE A REVIEW SESSION 6–7PMWED 04 FEB 2009 in Williams 103Conservative Forces, Work, and Potential Energy! W = F r( )"drWork Done (W) isIntegral of Force (F)! U = "WPotential Energy (U)is Negative of WorkDone! F r( )= "dUdrHence Force isNegative Derivativeof Potential EnergyCoulomb’s Law for Point ChargeP1P2q1q2P1P2q2! F12=kq1q2r2! F12= "dU12drForce [N]= Newton! U12=kq1q2r! U12= " F12dr#PotentialEnergy[J]=JouleElectricPotential[J/C]=[V]=Volt! V12=kq2r! V12= " E12dr# ! ! E 12=kq2r2ElectricField [N/C]=[V/m]! E12= "dV12drElectric Potential of a Point ChargeElectric Potential of a Point Charge2fPiRRV E ds E dskQ kQ kQdrr r R!!!= " # = " == " = + = +$ $$!!Note: if Q were anegative charge,V would be negativeElectric Potential of Many Point ChargesElectric Potential of Many Point Charges• Electric potential is aSCALAR not a vector.• Just calculate thepotential due to eachindividual point charge, andadd together! (Make sureyou get the SIGNScorrect!)q1q5q4q3q2!=iiirqkVr1r2r3r4r5PElectric Potential and ElectricElectric Potential and ElectricPotential EnergyPotential EnergyappU W q V! = = !+Q–QaWhat is the potential energy of a dipole?+Q• First: Bring charge +Q: no work involved, no potential energy.• The charge +Q has created an electric potential everywhere, V(r)!= kQ/r –Qa• Second: The work needed to bring the charge –Q to a distance a fromthe charge +Q is Wapp!=!U = (-Q)V = (–Q)(+kQ/a) = -kQ2/a• The dipole has a negative potential energy equal to -kQ2/a: we had todo negative work to build the dipole (electric field did positive work).Positive WorkPositive WorkNegative WorkNegative Work+Q+Qa+Q–QaPotential Energy of A System of ChargesPotential Energy of A System of Charges• 4 point charges (each +Q andequal mass) are connected bystrings, forming a square ofside L• If all four strings suddenlysnap, what is the kineticenergy of each charge whenthey are very far apart?• Use conservation of energy:– Final kinetic energy of all fourcharges = initial potential energystored = energy required toassemble the system of charges+Q+Q+Q+QDo this from scratch!Potential Energy of A System ofPotential Energy of A System ofCharges: SolutionCharges: Solution• No energy needed to bringin first charge: U1=022 1kQU QVL= =• Energy needed to bring in2nd charge:2 23 1 2( )2kQ kQU QV Q V VLL= = + = +• Energy needed to bring in3rd charge =• Energy needed to bring in4th charge =+Q+Q+Q+QTotal potential energy is sum ofall the individual terms shown onleft hand side =2 24 1 2 32( )2kQ kQU QV Q V V VLL= = + + = +! kQ2L4 + 2( ) So, final kinetic energy of eachcharge =! kQ24L4 + 2( ) L! 2LExampleExamplePositive and negative charges of equal magnitude Qare held in a circle of radius R.1. What is the electric potential at the center ofeach circle?• VA =• VB =• VC = 2. Draw an arrow representing the approximatedirection of the electric field at the center of eachcircle.3. Which system has the highest electric potentialenergy?–Q+QACB! k +3Q " 2Q( )/ r = +kQ / r!=iiirqkV! k +2Q " 4Q( )/ r = "2kQ / rk +2Q ! 2Q( )/ r = 0UB =Q•Vb has largest un-canceled chargeElectric Potential of a Dipole (on axis)Electric Potential of a Dipole (on axis)( ) ( )2 2Q QV k ka ar r= !! +What is V at a point at an axial distance r away from themidpoint of a dipole (on side of positive charge)?ar–Q+Q( ) ( )2 2( )( )2 2a ar rkQa ar r! "+ # #$ %=$ %$ %# +& ')4(4220arQa!="#204 rp!"=Far away, when r >> a:VpElectric Potential on PerpendicularElectric Potential on PerpendicularBisector of DipoleBisector of DipoleYou bring a charge of Qo =–3C from infinity to a point Pon the perpendicular bisectorof a dipole as shown. Is thework that you do:a) Positive?b) Negative?c) Zero?a-Q+Q–3CPU!= QoV!=!Qo(–Q/d+Q/d)!=!0!dSummary:Summary:• Electric potential: work needed to bring +1C from infinity;units [V] = Volt• Electric potential uniquely defined for every point in space --independent of path!• Electric potential is a scalar — add contributions fromindividual point charges• We calculated the electric potential produced by a singlecharge: V!=!kq/r, and by continuous charge distributions :V!=!∫ kdq/r• Electric potential energy: work used to build the system,charge by charge. Use W!=!qV for each
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