Physics for Scientists & Engineers 2

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1 Equpotental Surfaces and Lnes Physcs for Scentsts & Engneers 2 Sprng Semester 2005 Lecture 9 January 25, 2005 Physcs for Scentsts&Engneers 2 1 When an electrc feld s present, the electrc potental has a gven value everywhere n space Ponts close together n space form an equpotental surface Charged partcles can move along equpotental surfaces wthout havng any work done on them by the electrc feld Equpotental surfaces exst n three dmensons We wll often take advantage of symmetres n the electrc potental and represent the equpotental surfaces as equpotental lnes n a plane Equpotental surface from eght pont charges fxed at the corners of a cube January 25, 2005 Physcs for Scentsts&Engneers 2 2 General Consderatons Electrc charges can move perpendcular to electrc feld lnes wthout have any work done on them by the electrc feld because the scalar product of the electrc feld and the dsplacement s zero If the work done by the electrc feld s zero, then the electrc potental must be constant V = " W e q = 0 # V s constant Thus equpotental surfaces and lnes must always be perpendcular to the electrc feld lnes Constant Electrc Feld A constant electrc feld has straght, evenly space electrc feld lnes The equpotental surfaces n the case of a constant electrc feld are equally spaced planes n three dmensons or equally spaced lnes n two dmensons January 25, 2005 Physcs for Scentsts&Engneers 2 January 25, 2005 Physcs for Scentsts&Engneers 2 4 1

2 Electrc Feld from a Sngle Pont Charge We have shown that the electrc feld lnes from a sngle pont charge are radal lnes emanatng from the pont charge The equpotental surfaces for a pont charge are concentrc spheres n three dmensons and concentrc crcles n two dmensons Electrc Feld from Two Oppostely Charged Pont Charges The electrc feld lnes from two oppostely charge pont charges are more complcated The electrc feld lnes orgnate on the postve charge and termnate on the negatve charge The equpotental lnes are always perpendcular to the electrc feld lnes The red lnes represent postve electrc potental The blue lnes represent negatve electrc potental Close to each charge, the equpotental lnes resemble those from a pont charge January 25, 2005 Physcs for Scentsts&Engneers 2 5 January 25, 2005 Physcs for Scentsts&Engneers 2 6 Electrc Feld from Two Identcal Pont Charges The electrc feld lnes from two dentcal pont charges are also complcated The electrc feld lnes orgnate on the postve charge and termnate on negatve charge at nfnty Agan, the equpotental lnes are always perpendcular to the electrc feld lnes There are only postve potentals Close to each charge, the equpotental lnes resemble those from a pont charge Calculatng the Potental from the Feld To calculate the electrc potental from the electrc feld we start wth the defnton of the work dw done on a partcle wth charge q by a force F over a dsplacement ds dw = Fd s In ths case the force s provded by the electrc feld F = qe dw = q Ed s Integratng the work done by the electrc force on the partcle as t moves n the electrc feld from some ntal pont to some fnal pont f we obtan W = f q Ed s January 25, 2005 Physcs for Scentsts&Engneers 2 7 January 25, 2005 Physcs for Scentsts&Engneers 2 8 2

3 Calculatng the Potental from the Feld (2) Rememberng the relaton between the change n electrc potental and the work done V = " W e q We get V = V f " V = " W e q = " f # Ed s Takng the conventon that the electrc potental s zero at nfnty we can express the electrc potental n terms of the electrc feld as V = Ed s " Electrc Potental of a Pont Charge We defne the electrc potental of a pont charge q n terms of the change n electrc potental requred to brng a postve test charge to a dstance R from nfnty n the presence of the electrc feld generated by the pont charge. Remember that the electrc feld from a pont charge q at a dstance r s gven by E = kq r 2 The drecton of the electrc feld from a pont charge s always radal. Assumng that we ntegrate from a dstance R from the pont charge along a radal to nfnty we obtan V = Ed s kq " = R " r dr = # kq $ ' R 2 % & r ( ) = kq R R January 25, 2005 Physcs for Scentsts&Engneers 2 9 January 25, 2005 Physcs for Scentsts&Engneers 2 10 Electrc Potental of a Pont Charge (2) Electrc Potental from a System of Charges The electrc potental V from a pont charge q at a dstance r s then V = kq r Negatve pont charge Postve pont charge We can calculate the electrc potental from a system of n pont charges by addng the potental from each charge at each pont n space n V = V = =1 n =1 kq r Ths summaton produces an electrc potental at all ponts n space that has a value but no drecton Calculatng the electrc potental from a group of pont charges s usually much smpler than calculatng the electrc feld January 25, 2005 Physcs for Scentsts&Engneers 2 11 January 25, 2005 Physcs for Scentsts&Engneers 2 12

4 Example - Superposton of Electrc Potental Assume we have a system of three pont charges: q 1 = µc q 2 = µc q = -.50 µc. q 1 s located at (0,a) q 2 s located at (0,0) q s located at (b,0) a = 8.00 m and b = 6.00 m. What s the electrc potental at pont P located at (b,a)? Example - Superposton of Electrc Potental (2) The electrc potental at pont P s gven by the sum of the electrc potental from the three charges kq V = " = k q 1 + q 2 + q % =1 r # $ r 1 r 2 r & ' = k " q 1 b + q 2 a 2 + b + q % # $ 2 a & ' # V = N/C % $ % ( ) "6 C V = 562 V ( ) 2 + ( 6.00 m) 6.00 m "6 C 8.00 m " m "6 C & ( ' ( January 25, 2005 Physcs for Scentsts&Engneers 2 1 January 25, 2005 Physcs for Scentsts&Engneers 2 14 Calculatng the Feld from the Potental We can calculate the electrc feld from the electrc potental startng wth V = W e," q dw = q Ed s Whch allows us to wrte qdv = q Ed s Ed s = dv If we look at the component of the electrc feld along the drecton of ds, we can wrte the magntude of the electrc feld as the partal dervatve E S "s Calculatng the Feld from the Potental (2) We can calculate any component of the electrc feld by takng the partal dervatve of the potental along the drecton of that component We can wrte the components of the electrc feld n terms of partal dervatves of the potental as E x "x ; E y "y ; E z "z In terms of graphcal representatons of the electrc potental, we can get an approxmate value for the electrc feld by measurng the gradent of the potental perpendcular to an equpotental lne January 25, 2005 Physcs for Scentsts&Engneers 2 15 January 25, 2005 Physcs for Scentsts&Engneers

5 Example - Graphcal Extracton of the Feld from the Potental Assume a system of three pont charges q 1 = 6.00 µc q 2 =.00 µc q = µc ( x 1, y 1 ) = ( 1.5 cm,9.0 cm) ( x 2, y 2 ) = ( 6.0 cm,8.0 cm) ( x, y ) = ( 5. cm,2.0 cm) Example - Graphcal Extracton of the Feld from the Potental (2) We calculate the magntude of the electrc feld at pont P To perform ths task, we draw a lne through pont P perpendcular to the equpotental lne reachng from the equpotental lne of 0 V to the lne of 2000V The length of ths lne s 1.5 cm. So the magntude of the electrc feld can be approxmated as E S "s = ( V) ( 0 V) = 1.#10 5 V/m 1.5 cm The drecton of the electrc feld ponts from the postve equpotental lne to the negatve potental lne January 25, 2005 Physcs for Scentsts&Engneers 2 17 January 25, 2005 Physcs for Scentsts&Engneers