Physcs 2113 Jonathan Dowlng Physcs 2113 Lecture 14: WED 18 FEB Electrc Potental II Danger!
Electrc Potental Energy, Unts : Electrc Potental Potental Energy = U = [J] = Joules Electrc Potental = V = U/q = [J/C] = [Nm/C] = [V] = Volts Electrc Feld = E = [N/C] = [V/m] = Volts per meter F = qe (Force s charge tmes Feld) U = qv (Potental Energy s charge tmes Potental) Electron Volt = 1eV = Work Needed to Move an Electron Through a Potental Dfference of 1V: W = qδv = e x 1V = 1.60 10 19 C x 1J/C = 1.60 10 19 J
Electrc Potental Energy = Joules Electrc potental energy dfference ΔU between two ponts = work needed to move a charge between the two ponts: ΔU = U f U = W dw =! F d! s F = q 0! E f f! W = dw = F d s! f!! = q 0 E d s ΔU = U f U = W = q 0 f! E d! s
Electrc Potental Voltage = Volts = Joules/ Coulomb! Electrc potental voltage! dfference ΔV between two ponts = work per unt charge needed to move a charge between the two ponts: ΔV = V f V = W/q = ΔU/q dw =! F d! s! F = q 0! E f W = dw = f q 0! E d! s ΔV = V f V = W q 0 = f! E d! s
Equal-Potental = Equpotental Surfaces W Δ V = V f V = = E!! ds q The Electrc Feld s Tangent to the Feld Lnes Equpotental Surfaces are Perpendcular to Feld Lnes 0 f E!" E!" E!" E!" Work Is Needed to Move a Charge Along a Feld Lne. No Work Is Needed to Move a Charge Along an Equpotental Surface (Or Back to the Surface Where t Started). Electrc Feld Lnes Always Pont Towards Equpotental Surfaces Wth Lower Potental.
Electrc Feld Lnes and Equpotental Surfaces Why am I smlng? I m About to Be Struck by Lghtnng! http://www.cco.caltech.edu/~phys1/java/phys1/efeld/efeld.html
Conservatve Forces The potental dfference between two ponts s ndependent of the path taken to calculate t: electrc forces are conservatve. W ΔU Δ V = V f V = = = E!! ds q q 0 0 f
Electrc Potental of a Pont Charge f P!! V = E ds = E ds = R kq kq kq = dr = + = + 2 r r R R Brng magnary + test charge q 0 n from nfnty! Note: f q were a negatve charge, V would be negatve If q 0 and are both + charges as shown, s the Work needed to brng q 0 from to P + or?
Electrc Potental of Many Pont Charges Electrc potental s a SCALAR not a vector. q 4 Just calculate the potental due to each ndvdual pont charge, and add together! (Make sure you get the SIGNS correct!) q 5 r 4 r 5 r 3 r 1 P r 2 q 3 q 2 V = k q r q 1
3 D = 2d For speed set d = e = 1! (a) V P = +e d + +e 2d = 3 2 (b) V P = +e d + +e 2d = 3 2 (c) V P = +e d + +e 2d = 3 2 V P (a) = V P (b) = V P (c) e d = 3 / 2 e d = 3 / 2 e d = 3 / 2 No vectors! Just add wth sgn. One over dstance. Snce all charges same and all dstances same all potentals same.
ICPP: Postve and negatve charges of equal magntude Q are held n a crcle of radus r. V = k q r Q +Q 1. What s the electrc potental voltage at the center of each crcle? V A = ( ) /r = +kq/r k +3Q 2Q A V B = ( ) /r = 2kQ/r k +2Q 4Q V C = ( ) / r = 0 k +2Q 2Q 2. Draw an arrow representng the approxmate drecton of B the electrc feld at the center of each crcle. 3. Whch system has the hghest potental energy? C U A =+q 0 V A has largest postve un-canceled charge
Potental Energy of A System of Charges 4 pont charges (each +Q and equal mass) are connected by strngs, formng a square of sde L If all four strngs suddenly snap, what s the knetc energy of each charge when they are very far apart? +Q +Q Use conservaton of energy: Fnal knetc energy of all four charges = ntal potental energy stored = energy requred to assemble the system of charges +Q +Q If each charge has mass m, fnd the velocty of each charge long after the strng snaps. Let s do ths from scratch!
Potental Energy of A System of No energy needed to brng n frst charge: U 1 =0 Energy needed to brng n 2nd charge: U Charges: Soluton = QV = 2 1 Energy needed to brng n 3rd charge = kq U3 = QV = Q( V1 + V2 ) = + L kq L kq 2 2 2 Energy needed to brng n 4th charge = 2kQ U 4 = QV = Q( V1 + V2 + V3 ) = + L 2L kq 2 2 2L L +Q +Q 2L +Q +Q Total potental energy s sum of all the ndvdual terms shown on left hand sde = kq 2 ( ) L 4 + 2 So, fnal knetc energy of each charge =K=mv 2 /2 = kq 2 ( ) 4L 4 + 2
Potental Energy of a Dpole Δ U = Wapp = qδv +Q a Q What s the potental energy of a dpole? +Q Q a Frst: Brng charge +Q: no work nvolved, no potental energy. The charge +Q has created an electrc potental everywhere, V(r) = kq/r Second: The work needed to brng the charge Q to a dstance a from the charge +Q s W app = U = (-Q)V = ( Q)(+kQ/a) = -kq 2 /a The dpole has a negatve potental energy equal to -kq 2 /a: we had to do negatve work to buld the dpole (electrc feld dd postve work).
Electrc Potental of a Dpole (on axs) What s V at a pont at an axal dstance r away from the mdpont of a dpole (on sde of postve charge)? Q Q V = k k a a ( r ) ( r + ) 2 2 a a ( r + ) ( r ) kq 2 2 = a a ( r )( r + ) 2 2 Qa = 2 2 a 4πε 0( r ) 4 p a Q +Q r Far away, when r >> a: V = p 4πε r 0 2
IPPC: Electrc Potental on Perpendcular Bsector of Dpole You brng a charge of Q o = 3C from nfnty to a pont P on the perpendcular bsector of a dpole as shown. Is the work that you do: a) Postve? b) Negatve? c) Zero? a -Q +Q P d U = Q o V = Q o ( Q/d+Q/d) = 0 3C
4 P a V a = + V b = 0 V c = P b V a > V c > V b P c
Summary: Electrc potental: work needed to brng +1C from nfnty; unts V = Volt Electrc potental unquely defned for every pont n space -- ndependent of path! Electrc potental s a scalar add contrbutons from ndvdual pont charges We calculated the electrc potental produced by a sngle charge: V=kq/r, and by contnuous charge dstrbutons: dv=kdq/r Electrc potental energy: work used to buld the system, charge by charge. Use W=qV for each charge.
Mdterm Exam #1 AVG = 67; STDV=17 A: 90-100% B: 75-89% C: 60-74% D: 50-59% F: 49-0%