Physics 2102 Jonathan Dowling Physics 2102 Lecture 6 Electric Potential II Example qi V k ri i Positive and negative charges of equal magnitude Q are held in a circle of radius R 1 What is the electric potential at the center of each circle VA k 3Q 2Q r kQ r VB k 2Q 4Q r 2kQ r VC k 2Q 2Q r 0 2 Draw an arrow representing the approximate direction of the electric field at the center of each circle 3 Which system has the highest electric potential energy UB Q Q A B C Electric Potential of a Dipole on axis What is V at a point at an axial distance r away from the midpoint of a dipole on side of positive charge V k Q k Q a a r r 2 2 a a r 2 r 2 kQ a a r r 2 2 Qa 2 a 4 0 r 2 4 a Q Q r Far away when r a p V 4 0 r 2 Electric Potential on Perpendicular Bisector of Dipole You bring a charge of Qo 3C from infinity to a point P on the perpendicular bisector of a dipole as shown Is the work that you do a Positive b Negative c Zero U QoV Qo Q d Q d 0 a Q Q d P 3C Continuous Charge Distributions Divide the charge distribution into differential elements Write down an expression for potential from a typical element treat as point charge Integrate Simple example circular rod of radius r total charge Q find V at center r dq dq V k r k Q dq k r r Potential of Continuous Charge Distribution Example Uniformly charged rod Total charge Q Length L What is V at position P shown x P dx L a Q L dq dx L kdq k dx V r L a x 0 k ln L a x L 0 L a V k ln a Electric Field Potential A Simple Relationship Focus only on a simple case electric field that points along x axis but whose magnitude varies with x Notice the following Point charge E kQ r2 V kQ r dV Ex dx Dipole far away E kp r3 V kp r2 E is given by a DERIVATIVE of V f r r Of course V E ds i Note MINUS sign Units for E VOLTS METER V m Electric Field Potential Example Hollow metal sphere of radius R has a charge q Which of the following is the electric potential V as a function of distance r from center of sphere V 1 r a V r r R V q c r R b r R 1 r r 1 r r Electric Field Potential Example q E V Outside the sphere Replace by point charge Inside the sphere E 0 Gauss Law E dV dr 0 IFF V constant 1 2 r 1 r dV E dr d Q k dr r Q k 2 r Equipotentials and Conductors Conducting surfaces are EQUIPOTENTIALs At surface of conductor E is normal to surface Hence no work needed to move a charge from one point on a conductor surface to another Equipotentials are normal to E so they follow the shape of the conductor near the surface Conductors change the field around them An uncharged conductor A uniform electric field An uncharged conductor in the initially uniform electric field Sharp conductors Charge density is higher at conductor surfaces that have small radius of curvature E 0 for a conductor hence STRONGER electric fields at sharply curved surfaces Used for attracting or getting rid of charge lightning rods Van de Graaf metal brush transfers charge from rubber belt Mars pathfinder mission tungsten points used to get rid of accumulated charge on rover electric breakdown on Mars occurs at 100 V m NASA 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 point charges 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 by charge 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 of conductors
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