Electric Potential IIElectric Potential IIPhysics 2102Jonathan DowlingPhysics 2102Physics 2102Lecture 6Lecture 6ExampleExamplePositive and negative charges of equalmagnitude Q are held in a circle of radius R.1. What is the electric potential at the centerof each circle?• VA =• VB =• VC = 2. Draw an arrow representing theapproximate direction of the electric field atthe center of each circle.3. Which system has the highest electricpotential energy?–Q+QACB k +3Q ! 2Q( )/ r = +kQ / r!=iiirqkV k +2Q ! 4Q( )/ r = !2kQ / rk +2Q ! 2Q( )/ r = 0UBElectric 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:VElectric Potential on PerpendicularElectric Potential on PerpendicularBisector of DipoleBisector of DipoleYou bring a charge of Qo = –3Cfrom infinity to a point P on theperpendicular bisector of a dipoleas shown. Is the work that you do:a) Positive?b) Negative?c) Zero?a-Q+Q-3CPU= QoV=Qo(–Q/d+Q/d)=0 dContinuous ChargeContinuous ChargeDistributionsDistributions• Divide the chargedistribution intodifferential elements• Write down an expressionfor potential from atypical element — treat aspoint charge• Integrate!• Simple example: circularrod of radius r, totalcharge Q; find V at center.dqV kr=!dqrk Qdq kr r= =!Potential of Continuous ChargePotential of Continuous ChargeDistribution: ExampleDistribution: Example/Q L!=dxdq!=!!"+==LxaLdxkrkdqV0)(#[ ]LxaLk0)ln( !+!="!"#$%&+=aaLkV ln'• Uniformly charged rod• Total charge Q• Length L• What is V at position Pshown?PxLadxElectric Field & Potential:Electric Field & Potential:A Simple Relationship!A Simple Relationship!Notice the following:• Point charge:– E = kQ/r2– V = kQ/r• Dipole (far away):– E ~ kp/r3– V ~ kp/r2• E is given by a DERIVATIVEof V!• Of course!dxdVEx!=Focus only on a simple case:electric field that pointsalong +x axis but whosemagnitude varies with x.Note:• MINUS sign!• Units for E --VOLTS/METER (V/m)fiV E ds! = " •#rrElectric Field & Potential: ExampleElectric Field & Potential: Example• Hollow metal sphere ofradius R has a charge +q• Which of the following isthe electric potential V asa function of distance rfrom center of sphere?+qVr1!rr=R(a)Vr1!rr=R(c)Vr1!rr=R(b)+qOutside the sphere:• Replace by point charge!Inside the sphere:• E =0 (Gauss’ Law) • E = –dV/dr = 0 IFF V=constant2dVEdrd Qkdr rQkr= !" #= !$ %& '=Vr1!Electric Field & Potential: ExampleElectric Field & Potential: ExampleE21r!Equipotentials Equipotentials and Conductorsand Conductors• Conducting surfaces areEQUIPOTENTIALs• At surface of conductor, E isnormal to surface• Hence, no work needed to move acharge from one point on aconductor surface to another• Equipotentials are normal to E, sothey follow the shape of theconductor near the surface.Conductors change the fieldConductors change the fieldaround them!around them!An uncharged conductor:A uniform electric field:An uncharged conductor in theinitially uniform electric field:““SharpSharp””conductorsconductors• Charge density is higher atconductor surfaces that havesmall radius of curvature• E = σ/ε0 for a conductor, henceSTRONGER electric fields atsharply curved surfaces!• Used for attracting or getting ridof charge:– lightning rods– Van de Graaf -- metal brushtransfers charge from rubber belt– Mars pathfinder mission --tungsten points used to get rid ofaccumulated charge on rover(electric breakdown on Marsoccurs at ~100 V/m)(NASA)Summary:Summary:• Electric potential: work needed to bring +1C from infinity; units = V• Electric potential uniquely defined for every point in space --independent of path!• Electric potential is a scalar -- add contributions from individual pointcharges• We calculated the electric potential produced by a single charge:V=kq/r, and by continuous charge distributions : V=∫ kdq/r• Electric field and electric potential: E= −dV/dx• Electric potential energy: work used to build the system, charge bycharge. Use W=qV for each charge.• Conductors: the charges move to make their surface equipotentials.• Charge density and electric field are higher on sharp points
View Full Document
Unlocking...