Review of Potential Energy. The Electric Potential. Plotting Fields and Potentials. Electric Potential of a Point Charge

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Duane McDaniel
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1 eview of Potential negy Potential enegy U() can be used to descibe a consevative foce. efeence point (U) can be chosen fo convenience. Wok done by F : W F d s F d (1D) Change in P.. : U U f U i W Foce fom P.. : F du d (moe igoously, F U ) Wok is independent of the path. U is unchanged if we etun to the same point. Fo consevative foces, K + U is constant without nonconsevative foces i U f 1 The lectic Potential Just like we defined as the foce pe unit chage, define the electic potential V() as the potential enegy pe unit chage. V U q F q U F d s V and F ae vectos. V and U ae scalas. V ; y V y ; V Altenate units fo : Volts/mete (same as Newtons/Coulomb) d s V is measued in volts. Moving 1C though 1V potential diffeence does 1J of wok. 1V 1 J C 1 Nm C 2 Plotting Fields and Potentials Given V(), plot (). V Changing V by a constant leaves unchanged. Note: V must be continuous need not be, and often changes discontinuously at sufaces V 2 3 lectic Potential of a Point Chage The efeence point fo the electic potential is often placed at infinity. V( ) + ds Integate the field fom infinity to get V() fo a point chage. V () d s ()d d 2 V () 4 1

2 Potential negy of a Chage in a Unifom lectic Field What is the change in potential enegy of a 3C chage in a unifom electic field of 1 N/C if it is moved along the path shown? 5m U dw F d s q 1 N () C 3C () 5m 15Nm 15J lectic Potential of a Chage in a Unifom Field What is the change in the electic potential between two points in a unifom electic field of 1 N/C sepaated by 5m along the field? 5m V d s () 1 N C 5m Nm 5 C 5 J C 5V The potential enegy gets lowe. If a positive chage wee eleased in this field it would acceleate to the ight. efeence point (U) is abitay hee. U We don t need chages in a egion fo thee to be a potential diffeence. Voltage efeence point (V) is abitay just like U. V 5 6 quipotential Sufaces and the lectic Field quipotential sufaces ae sufaces of constant V. quipotentials ae pependicula to the field lines. -q Scalas vs. Vectos Which of the following aangements of chages have eo electic field at thei midpoint? Which have eo potential? 4V 3V 2V 1V V -1V -2V -3V -4V UqV Positive chages move to lowe V when allowed to move feely. Negative chages move to highe V. Both ae moving to lowe U! A B C D 7 8 2

3 ank the Potentials Point chages ae placed along the cicumfeences of thee identical cicles. ank the potentials at the cente of the cicles. Potential is a scala. All chages ae equidistant fom the cente and equal in magnitude. ach contibutes +V o -V to the potential. A: +3V-2V+V B: +2V-4V-2V C: +2V-2VV A,C,B -q A B C 9 Potential negy of a System of Chages Fou equal point chages ae aanged in a squae. If the chages ae eleased, what is the kinetic enegy of each chage when they ae vey fa apat? Use consevation of enegy. When chages ae fa apat, U will be eo. Initially tot U Finally tot K By symmety, U and K ae equally shaed among the fou chages. Waning: Calculating Σq i V i double counts the potential enegy. You need to build up the chage distibution step-by-step. 1 Building the Squae U W by W against Fist chage can be placed without doing any wok. W 1 W 2 W W W tot W 1 +W 2 +W 3 +W 4 q q πε ach chage has a final kinetic enegy of W tot /4 11 Wok Done by the lectic Field How much wok is done by the electic field when a chage of -5 C is moved fom a V potential to 1V? epeat the calculation fo +5C. W U q V qv ( f V i ) ( 5C)(1V V ) 5J Field does positive wok on the negative chage when going to highe electic potential -5C F q q V +5C V 1V W U q V qv ( f V i ) (5C)(1V V) 5J This motion will not happen unless anothe foce is involved in moving the chage (o it has an initial velocity) 12 3

4 Acceleating Chages with lectic Fields What is the kinetic enegy of an electon which begins at est and taveses a potential diffeence of +1V? C V +1V K U q V qv ( f V i ) ( C)(1V V ) J 1 ev (electon - volt) 1J ev lecton volts ae commonly used in discussing enegies of electons and othe elementay paticles. Conventional television pictue tubes use potentials of tens of thousands of volts to acceleate electons. 13 Beakdown in Ai and the Wimshust Machine Dy ai is a poo conducto because thee ae few fee chages. When > V/m, some electons begin to be pulled fom nitogen molecules, ceating fee electons and positive ions. These electons acceleate and knock loose moe electons beakdown See acs, lightning-like dischages, etc. At what potential can you make a 5 cm spak in dy ai? V d 3 V 16 m V.5m 3 1 V 6 m V d + - (Thee ae ticks to lowe the equied voltage.) 14 Potential of an lectic Dipole (on-ais) What is the potential of a dipole on its ais? q q V + V d + d 2 2 V () V + +V q + d 1 1 d + d q 2 d 2 4πε d d 2 qd 1 p 2 d fo >> d P -q 15 d amples of Potentials and Fields Point chage : V V Dipole (fa away, along ais) : p V 2 V inealy changing potential : V K 2 p 2πε 3 V K σ ε -q d 16 4

5 Continuous Chage Distibutions Potential of a Cut ing Calculating potentials fo continuous chage distibutions is much like calculating fields, only simple. V is a scala, so thee is only one integal to do and no components to woy about. Divide chage into small pieces dq. Teat each as a point chage. Integate ove the object. V at cente of ing with chage : dq λdφ λ 2π 2π dq V 17 1 λ dq dφ 1 2π (2π) dq What is the potential due to a patial ing of chage coveing a 3 angle? The total chage is -5. Compute the potential on the ais of the ing. The ing has a 6 gap, so it is 5/6 complete. It can be modeled as a supeposition of a -6 complete ing and a +1 segment spanning 6. All chage is the same distance fom a point on the -ais. k( 6) V k Building the Squae U W by W against Fist chage can be placed without doing any wok. Building a chaged sphee Build up a hollow sphee of chage by adding chage dq at adius. W 1 W 2 W W W tot W 1 +W 2 +W 3 +W 4 q q πε ach chage has a final kinetic enegy of W tot /4 +dq

6 Adding Chage to a Sphee Build up a hollow sphee of chage by adding chage dq at adius. q du Vdq dq q U q dq 1 1 U 2 8πε 2 q2 Self enegy of suface chage +dq 21 Potential fo a Hollow Sphee What is the potential fo a hollow spheical shell of adius with chage +? Outside-same as a point chage + at the oigin. V () > Inside- so V is constant. V () < V 1/ 2 1/ 22 Potential of a Finite ine of Chage (along the line) - a What is the potential of a line of chage of length along the line of the chage? a 1) Place oigin at end of line. is coodinate along line. 2a) dq λd λ 2b) a - 3) dv dq λd (a ) 4) Integate fom - to [ ] V /2 d log(a ) /2 (a ) log a + a 23 dl y Potential of a ing of Chage ϕ Chage : dq λd d dϕ λ Distance : dq dv 1 V dv λ 2ε λdϕ π λ dϕ π 24 6

7 X-/2 Potential of a Finite ine of Chage What is the potential of a line of chage of length and chage along its pependicula bisecto? d P y X+/2 1) Place oigin at midpoint of line. is coodinate along line, and y is pependicula coodinate. 2a) dq λd λ 2b) 2 + y 2 3) dv dq λd 2 + y 2 4) Integate fom -/2 to /2 V /2 /2 d 2 + y 2 log( y 2 ) /2 /2 Calculating the Field fom the Potential V ; y V y ; V Along a paticula path, the potential diffeence is the integal of the electic field components along the path. V d s ; dv - d s s V s "diectional deivative" s Potential of a Chaged Disk P d y Aea of ing fom to + d is 2π d ( ) σ dq σ 2π d π 2 Distance fom ing to field point is dq dv 2πσ d V σ d σ ε ε σ σ ε ε 1 2 >> σ ε σ2 4ε 27 Potential of a Chaged Disk P d y >> V σ ε σ ε 2 σ σ 2ε 2ε Fist tem is a constant. Can subtact fom V. σ V 2ε (This gives ight field fo infinite sheet) 28 7

Potential of an Infinite Hollow Cylinde Use Gauss aw to get. () V so V () - () d () λ 2πε > () < λ 1 V() - d 2πε λ log() 2πε V() λ log() 2πε > < λ V as only if chages ae confined in a finite volume.

8 Potential of an Infinite Hollow Cylinde Use Gauss aw to get. () V so V () - () d () λ 2πε > () < λ 1 V() - d 2πε λ log() 2πε V() λ log() 2πε > < λ V as only if chages ae confined in a finite volume. 29 quipotentials and Conductos fo conductos in electostatic equilibium. Conductos ae equipotentials No wok is needed to move a chage within o along suface of a conducto. quipotentials must follow the suface of the conducto when neaby, and they ae nomal to field is nomal to conducto sufaces. 3 Chage Density on Metal Sufaces Two chaged metal sphees have 2 diffeent adii. 1 If they ae connected by a conducting wie, what is the atio of thei chages? Once connected, they ae an equipotential. What about σ (and )? 1 V 1 V 2 ( ) σ ε σ 1 1 4π 2 1 σ 2 1 4π(1) π 2 1 In geneal, on a given conducto is invesely popotional to the local adius of cuvatue. Shap points have high fields! Potential Baie A positon is moving at m/s in the positive diection of an ais when, at, it encountes an electic field along the ais. The electic potential associated with the field is given in the figue. Take V S 5 V. a) Does the positon emege fom the field at (its motion evesed) o at.5 m (motion not evesed). b)what is its speed when it emeges? σ 1 1σ

9 Consevation of Cuent Dift Velocity lectons don t pile up, they keep flowing, just like wate in a pipe. i 1 i 2 + i 3 i 1 i 2 Conductos conduct because some of the electons ae not bound to individual atoms. These electons ae in constant fast andom motion, and collide fequently with vibating nuclei. Cuent is conseved. i 3 33 When an electic field is applied to a conducto, electons dift in the diection opposite the field. This small dift, supeposed on the andom motion, esults in cuent. +V I v e 34 A model of dift velocity Cuent Density and Dift Velocity v d aτ F m τ e m τ Also: J env d v d J en How ae cuent density and dift velocity elated? Imagine a cylinde oiented paallel to v d. v d Solve fo J: Combine: J e 2 nτ m J en eτ m and J ae popotional to each othe. Aside: Vaiables in backets ae ~independent of. τ Does not depend on since v d << vavg 35 n fee electons pe unit volume na fee electons in the cylinde fee chage in cylinde qena How long does the chage take to leave the cylinde? It needs to tavel a distance v d t so t/v d J q At ena A v d env d J envd 36 9

10 Dift Velocity in Coppe Conductivity and esistivity What is the dift velocity v d of the conduction electons in a coppe wie of 1 mm adius caying a unifom cuent of 2 ma? J env d I A Assume 1 conduction electon pe atom n 1electon # Cu Cu volume 1electon # Cu kg Cu kg volume 1electon 1Cu 1amu kg Cu 64 amu kg m 3 81 electons 28 m 3 v d J en I ena 3 I enπ 21 A 2 π(1 3 m) m 3 ( C) m 51 7 s Jσ ρ 1/σ Combine: σ is conductivity ρ is esistivity ρj elating ρ and σ to τ Time fo cuent flow J e 2 nτ m σ e 2 nτ m σ e 2 nτ m How long does it take electons to get fom the ca battey to the stating moto? Assume a cuent of 3 A and the electons tavel though Cu wie with coss-sectional aea.21 cm 2 and length.85 m. The numbe of caies pe unit volume is m

Why do cuents change faste than the electons dift? The dift velocity is ~51-7 m/s, o 1.8 mm/hou! Why doesn t it take seveal days to tun on a light bulb with a switch?

11 Why do cuents change faste than the electons dift? The dift velocity is ~51-7 m/s, o 1.8 mm/hou! Why doesn t it take seveal days to tun on a light bulb with a switch? The electic field appeas eveywhee in the wie immediately. The only delays ae associated with the speed of light. lectons eveywhee in the wie begin difting simultaneously. Individual electons don t need to tavese the whole wie. Those leaving one end ae eplaced fom the othe end. Just like wate in a pipe Cuent flow The figue shows a 4. cm wide plastic film being wapped onto a 2. cm diamete olle that tuns at 9 pm. The plastic has a unifom suface chage density of -2. nc/ cm 2. a.what is the cuent in the moving film? b.how long does it take the olle to accumulate a chage of -1 µc?

Potential Energy. The change U in the potential energy. is defined to equal to the negative of the work. done by a conservative force

Potential Energy. The change U in the potential energy. is defined to equal to the negative of the work. done by a conservative force Potential negy The change U in the potential enegy is defined to equal to the negative of the wok done by a consevative foce duing the shift fom an initial to a final state. U = U U = W F c = F c d Potential