AP Physics lectic Potential negy Review of some vital peviously coveed mateial. The impotance of the ealie concepts will be made clea as we poceed. Wok takes place when a foce acts ove a distance. W F The unit fo wok is the joule (J). A joule is a newton mete. Potential enegy fom gavity is g mgh. The foce of gavity acting on an object with mass is always diected towads the cente of the eath. Remembe? When you lift an object you do wok on it. As its height is inceased, its potential enegy inceases. The wok that you do lifting it is eual to the incease in potential enegy of the thing. Imagine a skyscape. An object that is on the 75 floo has moe potential enegy than does an object on the nd floo, tue? F g incease in gavitational potential enegy If we move an object fom the th floo to the th floo, we ve (a) done wok on it and (b) inceased its potential enegy. If we move an identical object fom the 5 th floo to the 35 th floo, we ve (a) done wok on it and (b) inceased its potential enegy. In fact we ve done the same amount of wok on both occasions and we ve inceased the potential enegy of the object by the same amount, tue? We ve done ten floos woth of wok and inceased the potential enegy ten floos woth. We could make it even moe geneal by looking at potential enegy changes pe kilogam. Then we wouldn t have to woy about the objects all having the same mass. We would be dealing with a change in potential enegy pe kilogam. Analyzing changes in gavitational potential enegy doesn t tun out to be teibly useful in dealing with gavity, enegy, and wok. D D D Fo the same height change, potential enegy change is the same 3
Well, it isn t useful fo gavity fields, but it is extemely useful when dealing with electic fields. Looking at an electic field aound a chaged object we find that we also have wok being done and potential enegy changes being made when we move chaged objects about. We have a compaable situation to that of gavity when we have two opposite chages. We have a lage chage Q. Placed nea this chage is a small test chage. As we incease the distance between and Q, we have to do wok to move the chage and we incease its potential enegy. If the chages ae the same, then we have a diffeent situation. It takes wok to move the test chage close to the othe chage (the foce is epulsive between them). As the two chages move close togethe, the potential enegy inceases. F + F F + incease in electic potential enegy - - + + - - nlike chages Foce is attactive - - incease in electic potential enegy + + Like chages Foce is epulsive F lectic Potential negy: Let s go ahead and develop an euation fo the potential enegy of a chage in an electic field. Let s look a simple situation with two chages. Now, deiving an euation fo electical potential enegy sounds too had. We haven t studied anything about electical potential enegy, have we? Oh, ye, of little faith. We don t need to study anything special, we aleady know all that we need to in ode to figue this out. We know that potential enegy is foce times distance. F The foce acting between the electon and the poton is: F 4 We can plug in Coulomb s law fo the foce in the potential enegy euation. 4 4 33
So thee it is. An euation fo electical potential enegy: 4 An electon in a hydogen atom is at its lowest enegy state. It is at a distance of 5.9 x - m fom the poton in the nucleus. What is its potential enegy elative to the poton? We can look up the chages fo the paticle, although by now, you pobably know what they ae by heat. Anyway, hee ae the chages: 9 9.6 x C.6 x C 4 9 9 5.9 9 Nm 8.99 x.6 x C.6 x C C x m 8 4.35 x J Wok in the lectic Field: alie we mentioned that wok was done when a paticle was moved fom one point to anothe in an electic field. This is accompanied by an incease in the chage s potential enegy (the chage that gets moved). If the two chages ae unlike: Pull the chages apat incease potential enegy (just like with gavity). Push the chages togethe - potential enegy deceases. If the two chages ae the same: Pull the chages apat - decease the potential enegy. Push togethe - potential enegy inceases. The impotant thing hee is that the wok done in moving the chage is eual to the potential enegy diffeence between the two points The citical thing is the change in displacement the diffeence between the stating point and the final position of the chage. We don t cae about what the actual distance between the chages is. We know that wok is: W F The foce exeted on a test chage in an electic field is: 34
F F We now put these two euations togethe and we get: W Fd d d This gives us an euation fo the wok done in an electic field in moving a chage fom one position to anothe: W d W is the wok done in Joules, is the chage of the test chage (not the chage making the field), is the electic field stength, and d is the displacement of the chage. In the dawing to the ight, we have a unifom electic field of stength. A chage is moved fom point A to point B, which is a distance d. The wok needed to do this is given by the euation we just developed. A B The magnitude of the wok done in moving the chage is eual to the change in potential enegy. Theefoe we can wite: W d The negative sign simply means that the potential enegy will incease if the chage is negative o decease if the chage is positive. We ecall the euation fo wok: W d We plug this into the euation fo the change in potential enegy. d This is anothe useful euation. Potential Diffeence: We can make things even moe geneal. Instead of looking at the total potential enegy, we can look at the potential enegy pe Coulomb of chage. This, the potential enegy pe chage, is called potential diffeence. 35
Potential Diffeence Change in potential enegy pe Chage The symbol fo potential diffeence is o. Potential Diffeence If we look at the euation fo potential enegy and divide both sides by the chage, then we get the potential diffeence. d We divide both sides by (which is mathematically legal). d d This euation is commonly eaanged to this fom: d This is the euation that you will have available fo the AP Physics Test. Notice also that the electic field can be expessed in units of /m as well as N/C. Hee is the fomal definition fo potential diffeence between points a and b: If Potential diffeence is the change in potential enegy as a chage is moved fom a to b divided by the chage. d, then d d so On the AP Physics xam, the euations fo potential enegy ae combined togethe into one, looks like this: 4 And of couse, the wok is eual to the change in potential enegy, so 36
W 4 The unit fo potential diffeence is the volt (). The volt is a joule/coulomb. J C A potential diffeence of one volt means that it takes one joule of enegy to move a one Coulomb chage in the field. A potential diffeence of means that it would take J to move C of chage in the field. Potential diffeence is also called electic potential o voltage. Potential diffeence is one of the most impotant concepts in physics when you ae dealing with electicity, so you want to take special cae to maste the thing. It is essential that you have a solid undestanding about potential diffeence and how it woks. Chage moves because of the potential diffeence,. The bigge the potential diffeence between two points, the moe the chage will want to move. lectic potential is a scala uantity. Hee ae the euations that you will have to wok with on the AP Physics Test: F 4 Avg d We ll now make use of them to solve the odd poblem o two. The potential diffeence between two points is.. What amount of wok is needed to move a. C chage within the field? W 4 6 6 5 W. x C. 4. x J.4 x J Two chaged plates ae 5. cm apat. The electic field between them is 775 N/C. What is (a) the potential diffeence between the plates and (b) what wok is done moving an electon fom one plate to anothe? 37
N J d 775.5 m 38.8 38.8 d C C (a) (b) We know the electon s chage, so this is faily simple: 9 9 8 W.6 x C 38.8 6. x J 6. x J The voltage between two big chaged up paallel plates is -.. The plates ae.5 cm apat. What is the electic field between them? 3. J 48 N 4.8 x N d C.5 m C C Notice that the teache has abandoned showing how the units all wok out. This is because in the electic wold with all of its wacky units, it is just too much touble. Hee s what you do. You put eveything into standad units: metes, Newtons, joules, Coulombs, olts, &tc. Then you just put down the pope unit that the answe is supposed to have. A poton is eleased fom est in a unifom electic field, = 8. x 4 /m. It is diected along the x axis. It s displacement is.5 m in the diection of the field. (a) What is the change in potential diffeence? (b) What is the change in electical potential enegy? (c) What is its velocity afte it taveled the.5 m? (a) We can find the potential diffeence since we know the electic field stength and the displacement. 4 d 8. x.5 m 4. x 4 d m (b) The potential enegy is easy to find. We know the chage of a poton and we know the potential diffeence it is exposed to. 9 4 J 5.6 x C 4. x 6.4 x J C (c) The poton s potential enegy will be conveted into kinetic enegy as it moves though the.5 m. So we can set the kinetic enegy eual to the potential enegy and solve fo the paticle s velocity. mv v We can look up the mass of a poton. m 38
5 kg m x 7 v 6.4 s.67 x kg m 6 v 7.66 x.8 x s m s Potential Diffeence and Conductos: In a conducto electons, fo all pactical puposes, ae fee to move about evey which way but loose. This means that thee is no potential diffeence within a conducto fom one point to anothe. If one end of a wie has a potential diffeence of, the othe end will also have a potential of. We physics types say that all points in a conducto ae at an euipotential. When conductos ae joined togethe, the chage ends up being evenly distibuting thoughout the newly fomed conductive system. The new conducto system will also be at an euipotential. 39