V-1 of 10 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is a scala thing, called the voltage o the electic potential. lectic fields and voltages ae two diffeent ways to descibe the same thing. (Note on teminology: The tet book uses the tem "electic potential", but it is easy to confuse electic potential with "potential enegy", which is something diffeent. So I will use the tem "voltage" instead.) Voltage oveview The voltage at a point in empty space is a numbe (not a vecto) measued in units called volts (V). Nea a positive chage, the voltage is high. Fa fom a positive chage, the voltage is low. Voltage is a kind of "electical height". Voltage is to chage like height is to mass. It takes a lot of enegy to place a mass at a geat height. Likewise, it takes a lot of enegy to place a positive chage at a place whee the voltage is high. lowe voltage hee highe voltage hee Only changes in voltage V between two diffeent locations have physical significance. The zeo of voltage is abitay, in the same way that the zeo of height is abitay. We define V in 2 equivalent ways: U of q V = change in potential enegy of a test chage divided by the test chage q V V V d B A B A Fo constant -field, this integal simplifies to V ( = change in position) The electic field is elated to the voltage in this way: lectic field is the ate of change of voltage with position. -field is measued in units of N/C, which tun out to be the same as Univesity of Coloado at Boulde
V-2 of 10 volts pe mete (V/m). -fields points fom high voltage to low voltage. Whee thee is a big - field, the voltage is vaying apidly with distance. high voltage -field low voltage In ode to undestand these stange, abstact definitions of voltage, we must eview wok and potential enegy Wok and Potential negy (U) Definition of wok done by a foce: conside an object pulled o pushed by a constant foce F. While the foce is applied, the object moves though a displacement of = f - i. F ( i ) ( f ) Notice that the diection of displacement is not the same as the diection of the foce, in geneal. Wok done by a foce F = WF F F cos F (constant F) F = component of foce along the diection of displacement If the foce F vaies duing the displacement (o the displacement is not a staight line), then we must use the moe geneal definition of wok done by a foce WF F d Wok is not a vecto, but it does have a sign (+) o (-). Wok is positive, negative, o zeo, depending on the angle between the foce and the displacement. F < 90, W positive F = 90, W = 0 F > 90, W negative Univesity of Coloado at Boulde
V-3 of 10 Definition of Potential negy U: Associated with consevative foces, such as gavity and electostatic foce, thee is a kind of enegy of position called potential enegy. The change in potential enegy U of a system is defined to be the negative of the wok done by the "field foce", which is the wok done by an "etenal agent" opposing the field. U W et = W field This is best undestood with an eample: A book of mass v f = 0 m is lifted upwad a height h by an "etenal agent" (a hand which eets a foce to oppose the foce of gavity). The h g foce of gavity is the "field". In this case, the wok done m by the hand is W et = +mgh. The wok done by the field v i = 0 (gavity) is W field = mgh. The change in the potential Foces on book: enegy of the eath/book system is U = W et = W field = +mgh. The wok done by the etenal agent went into the inceased gavitational potential enegy of the book. (The initial and final velocities ae zeo, so thee was no incease in kinetic enegy.) A consevative foce is foce fo which the amount of wok done depends only on the initial and final positions, not on the path taken in between. Only in the case that the wok done by the field in independent of the path, does it make any sense to associate a change in enegy with a change in position. Potential enegy is a useful concept because (if thee is no fiction, no dissipation) K + U = 0 K + U = constant (no dissipation) (K = kinetic enegy = ½ m v 2 ) F et F gav = mg Voltage We define electostatic potential enegy (not to be confused with electostatic potential o voltage) in the same way as we defined gavitational potential enegy, with the elation U = W et = W field. Conside two paallel metal plates (a capacito) with equal and opposite chages on the plates which ceate a unifom electic field between the plates. The field will push a test chage +q towad the negative plate with a constant foce of magnitude F = q. (The situation is Univesity of Coloado at Boulde
V-4 of 10 much like a mass in a gavitational field, but with the electic foce eplacing gavity.) Now imagine gabbing the chage with tweezes (an etenal agent) and pushing the chage +q a displacement against the electic field towad the positive plate. By definition, the change in electostatic potential enegy of the chage is U Uf Ui Wet Wfield Ffield q I ecommend that you do not ty to get the signs fom the equations it's too easy to get confused. Get the sign of U by asking whethe the wok done by the etenal agent is positive o negative and apply U = +W et. If the etenal agent has to push to make the chage go in the diection it does not want to go, then U is positive, and the potential enegy U inceases. But if the etenal agent has to estain the chage, pulling in the diection opposite the motion, then U is negative, and the potential enegy U deceases. +q F = q ( f ) ( i ) hi P lo P If the -field is not constant, then the wok done involves an integal f i field field f f U U U W F d q d. i i Now we ae eady fo the definition of voltage diffeence between two points in space. Notice that the change in P of the test chage q is popotional to q, so the atio U/q is independent of q. Recall that electic field is defined as the foce pe chage: the voltage diffeence V as the change in P pe chage: F on q. Similaly, we define q V U q f i d, o U q V Univesity of Coloado at Boulde
V-5 of 10 Remembe that the -field always points fom high voltage to low voltage: V ( if = constant ) high voltage -field low voltage The high voltage people look down thei electic-field noses at the poo, low-voltage people. If, then V = ( ) Vf Vi 0, Vi V f To say that "the voltage at a point in space is V" means this: if a test chage q is placed at that point, the potential enegy of the test chage q (the wok equied to place the chage thee) is U = q V. If the chage is moved fom one place to anothe, the change in P is U q V. Only changes in P and changes in V ae physically meaningful. We ae fee set the zeo of P and V anywhee we like. Units of voltage = [V] enegy chage joule coulomb volt (V). 1 V = 1 J/C Voltage nea a point chage V =? Answe: V() = kq Notice that this fomula gives V = 0 at =. When dealing with point chages, we always set the zeo of voltage at =. V V nea (+) chage is lage and positive. Q V nea ( ) chage is lage and negative. Univesity of Coloado at Boulde V
V-6 of 10 Poof: d ( i ) ( f ) d d d, so we have V V V V( ) V() d f i 0 k Q 1 k Q V() d d kq 2 Done. Notice that kq V() = is positive o negative, depending on the sign of Q. Voltage due to seveal chages If we have seveal chages Q 1, Q 2, Q 3,, the voltage at a point nea the chages is V V V V... V kq i tot 1 2 3 i i i i o dq k Poof: V tot d (1 2...) d V1 V 2... Voltages add like numbes, not like vectos. What good is voltage? Much easie to wok with V's (scalas) than with 's (vectos). asy way to compute P. Voltage eample: Two identical positive chages ae some distance d apat. What is the voltage at point midway between the chages? What is the -field midway between the chages? How much wok is equied to place a chage +q at? V tot = V 1 + V 2 = d V =? =? Univesity of Coloado at Boulde
V-7 of 10 V V V tot 1 2 k Q k Q 2k Q 2k Q 4k Q d / 2 d / 2 d d d The -field is zeo between the chages (Since tot 1 2 = 0. Daw a pictue to see this!) The wok equied to bing a test chage +q fom fa away to the point is positive, since it is had to put a (+) chage nea two othe (+) chages. You have to push to get the +q in place. The wok done is W et = U = +q V, whee V Vfinal Vinitial V(at ) V(at ) = W et 4k q Q d 0 4k Q d Units of electon-volts (ev) The SI units of enegy is the joule (J). 1 joule = 1 newtonmete = 1Nm Anothe, non-si, unit of enegy is the electon-volt (ev), often used by chemists. The ev is a vey convenient unit of enegy to use when woking with the enegies of electons o potons. Fom the elation U = q V, we see that enegy has the units of chage voltage. If the chage q = 1 e = chage of the electon and V = 1 volt, then U = q V = 1 e 1V = a unit of enegy called an "ev". Notice that the name "ev" eminds you what the unit is: it's an "e" times a "V" = 1 e 1 volt. How many joules in an ev? 1 ev = 1 e 1V = (1.6 10-19 C)(1 V) = 1.6 10-19 J 1 ev = 1.6 10-19 J If q = e (o a multiple of e), it is easie to use units of ev instead of joules when computing (wok done) = (change in P). ample of use of ev. A poton, stating at est, "falls" fom the positive plate to the negative plate on a capacito. The voltage diffeence between the plates is V = 1000 V. What is the final K of the poton (just befoe it hits the negative plate)? As the poton falls, it loses P and gains K. q = +e V = 1000 V V = 0 V K U q V 1e 1000 V 1000 ev Univesity of Coloado at Boulde
V-8 of 10 "quipotential Lines" = constant voltage lines Given, we can compute V d (if constant). Notice that if the displacement is pependicula to the diection of, then V 0. A suface of constant V is one along whichv = 0. quipotential (constant voltage) lines ae always at ight angles to the electic field. equipotential lines Computing fom V Given, we can compute V d (if constant). Given V = V(, y, z), how do we get? Suppose and is vey small so that constant along, then V V d V d " is the ate of change of V": but dv d only if the -ais is the diection of. Suppose ˆ is along the -ais, but not necessaily along. V dv dv ˆ V, y d dy, etc Actually, we must take "patial deivative" V means hold y, z constant, while taking deivative w..t Univesity of Coloado at Boulde
V-9 of 10 V V V So we conclude that, y, z y z, and we can wite V V V ˆ ˆ ˆ ˆ ˆ ˆ y y z z y z y z This elation is usually witten in tems of the gadient opeato : V V V V ˆ yˆ zˆ y z The electic field is the negative of the gadient of V o gad V. A Metal object in equilibim is an equipotential Metal objects (conductos) in electostatic equilibium ae always equipotentials (V = constant eveywhee inside and on the suface). V = = 0 (since ) ai + + + + + + + + + + + + + V = 0 (since = 0 ) metal V = 0 V = constant Univesity of Coloado at Boulde
V-10 of 10 negy of a collection of chages Suppose thee is a collection of chages, say q 1, q 2, and q 3, located as shown with sepaations 12, 23, and 31. This configuation of chages has an electostatic potential enegy associated with it. The potential enegy U is the enegy equied to assemble the chages, stating fom infinity. y q 1 12 31 q 2 23 q 3 Imagine assembling the chage configuation by binging the chages, one at a time, fom infinity to thei final positions. Stat with empty space and bing in the fist chage q1 and placing it at 1. This costs no enegy: U 1 = 0. But now binging up the second chage q2 costs enegy kq1 kq1q2 U2 q2 V( 2) whee V is the voltage due to the fist chage, V : U2. Finally, to bing in the last chage q3, we must do wok U3 q3 V( 3) whee V is now the kq1 kq 2 voltage due to both q1 and q2, V : kq q kq q U 1 3 2 3 3. So the total 31 23 electostatic potential enegy of the chage configuation is 12 31 23 12 U tot kq q kq q kq q kq q 1 2 1 3 2 3 i j. The fomula 12 31 23 i, j ij U tot kq q i j woks fo any i, j ij numbe of chages. Univesity of Coloado at Boulde