1 Name St.No. - Date(YY/MM/DD) / / Section Goup# UNIT 21: ELECTRICAL AND GRAVITATIONAL POTENTIAL Appoximate time two 100-minute sessions OBJECTIVES I began to think of gavity extending to the ob of the moon, and... I deduced that the foces which keep the planets in thei obs must be ecipocally as the squaes of thei distances fom the centes about which they evolve: and theeby compaed the foce equisite to keep the moon in he ob with the foce of gavity at the suface of the eath, and found them to answe petty nealy. All this was in the two plague yeas of 1665 and 1666, fo in those days I was in the pime of my age fo invention, and minded mathematics and philosophy moe than at any time since. Isaac Newton 1. To undestand the similaities in the mathematics used to descibe gavitational and electical foces. 2. To eview the mathematical definition of wok and potential in a consevative foce field. 3. To undestand the definition of electical potential, o voltage, and its similaity to the concept of gavitational potential. 4. To lean how to detemine electic field lines fom equipotential sufaces and vice vesa. 5. To lean how to map equipotentials in a plane esulting fom two point chages and two line electodes.
2 Page 21-2 Wokshop Physics II Activity Guide SFU OVERVIEW 10 min The entepise of physics is concened ultimately with mathematically descibing the fundamental foces of natue. Natue offes us seveal fundamental foces, which include a stong foce that holds the nuclei of atoms togethe, a weak foce that helps us descibe cetain kinds of adioactive decay in the nucleus, the foce of gavity, and the electomagnetic foce. JIMMY! Wake up. z! Uhh...My eyelids wee subject to insumountable gavitational foce. Soy ma am. Two kinds of foce dominate ou eveyday eality the gavitational foce acting between masses and the Coulomb foce acting between electical chages. The gavitational foce allows us to descibe mathematically how objects nea the suface of the eath ae attacted towad the eath and how the moon evolves aound the eath and planets evolve aound the sun. The genius of Newton was to ealize that objects as divese as falling apples and evolving planets ae both moving unde the action of the same gavitational foce. Similaly, the Coulomb foce allows us to descibe how one chage "falls" towad anothe o how an electon obits a poton in a hydogen atom. and NSF. Modified at SFU by S. Johnson and N Albeding, 2005, 2009, 2014.
3 Wokshop Physics II: Unit 21 Gavitational & Electical Potential Page 21-3 Autho: Piscilla Laws "falling" electon attacted to poton The fact that both the Coulomb and the gavitational foces lead to objects falling and to objects obiting aound each othe suggests that these foces might have the same mathematical fom. In this unit we will exploe the mathematical symmety between electical and gavitational foces fo two easons. Fist, it is beautiful to behold the unity that natue offes us as we use the same type of mathematics to pedict the motion of planets and galaxies, the falling of objects, the flow of electons in cicuits, and the natue of the hydogen atom and of othe chemical elements. Second, what you have aleady leaned about the influence of the gavitational foce on a mass and the concept of potential enegy in a gavitational field can be applied to aid you undestanding of the foces on chaged paticles. Similaly, a "gavitational" Gauss' law can be used to find the foces on a small mass in the pesence of a lage spheically symmetic mass. We will intoduce the concept of electical potential diffeences, which ae analogous to gavitational potential enegy diffeences. An undestanding of electical potential diffeence, commonly called voltage, is essential to undestanding the electical cicuits which ae used in physics eseach and which dominate this age of electonic technology. The unit will culminate in the actual measuement of electical potential diffeences due to an electic field fom two "line" conductos that lie in a plane and the mathematical pediction of the potential diffeences using a deivation based on Gauss' law.
4 Page 21-4 Wokshop Physics II Activity Guide SFU SESSION ONE: ELECTRICAL AND GRAVITATIONAL FORCES 10 min Compaison of Electical and Gavitational Foces Let's stat ou discussion of this compaison with the familia expession of the Coulomb foce exeted on chage 2 by chage 1. q 1 unit vecto F 12 = k e q 1 q 2 2 ˆ 12 q 2 ˆ! 12 k e = N m2 C 2 Chales Coulomb did his expeimental investigations of this foce in the 18th centuy by exploing the foces between two small chaged sphees. Much late, in the 20th centuy, Coulomb's law enabled scientists to design cyclotons and othe types of acceleatos fo moving chaged paticles in cicula obits at high speeds. Newton's discovey of the univesal law of gavitation came the othe way aound. He thought about obits fist. This was back in the 17th centuy, long befoe Coulomb began his studies. A statement of Newton's univesal law of gavitation descibing the foce expeienced by mass 2 due to the pesence of mass 1 is shown below in moden mathematical notation: m 1 m 2 unit vecto ˆ! 12 F 12 = G m 1m 2 2 ˆ 12 G = N m2 kg 2 and NSF. Modified at SFU by S. Johnson and N Albeding, 2005, 2009, 2014.
5 Wokshop Physics II: Unit 21 Gavitational & Electical Potential Page 21-5 Autho: Piscilla Laws About the time that Coulomb did his expeiments with electical chages in the 18th centuy, one of his contempoaies, Heny Cavendish, did a diect expeiment to detemine the natue of the gavitational foce between two spheical masses in a laboatoy. This confimed Newton's gavitational foce law and allowed him to detemine the gavitational constant, G. A fact emeges that is quite amazing. Both types of foces, electical and gavitational, ae vey simila. Essentially the same mathematics can be used to descibe obital and linea motions due to eithe electical o gavitational inteactions of the tiniest fundamental paticles o the lagest galaxies. [This statement needs to be qualified a bit when electons, potons and othe fundamental paticles ae consideed. A new field of study called wave mechanics was developed in the ealy pat of the 20th centuy to take into account the wave natue of matte, which we don't actually study in intoductoy physics. Howeve, even in wave mechanical calculations electical foces like those shown above ae used.] Activity 21-1: The Electical vs. the Gavitational Foce Examine the mathematical expession fo the two foce laws. (a) What is the same about the two foce laws? (b) What is diffeent? Fo example, is the foce between two like masses attactive o epulsive? How about two like chages? What pat of each equation detemines whethe the like chages o masses ae attactive o epulsive? (c) Do you think negative mass could exist? If thee is negative mass, would two negative masses attact o epel?
6 Page 21-6 Wokshop Physics II Activity Guide SFU 15 min Which Foce is Stonge-- Electical o Gavitational? Gavitational foces hold the planets in ou sola system in obit and account fo the motions of matte in galaxies. Electical foces seve to hold atoms and molecules togethe. If we conside two of the most common fundamental paticles, the electon and the poton, how do thei electical and gavitational foces compae with each othe? Let's peek into the hydogen atom and compae the gavitational foce on the electon due to inteaction of its mass with that of the poton to the electical foce between the two paticles as a esult of thei chage. In ode to do the calculation you'll need to use some well known constants. Electon: me = kg qe = C Poton: mp = kg qp = C Distance between the electon and poton: = 1.0 x m Activity 21-2: The Electical vs. the Gavitational Foce in the Hydogen Atom (a) Calculate the magnitude of the electical foce on the electon. Is it attactive o epulsive? (b) Calculate the magnitude of the gavitational foce on the electon. I s it attactive o epulsive? (c) Which is lage? By what facto (i.e. what is the atio)? (d) Which foce ae you moe awae of on a daily basis? If you answe does not agee with that in pat (c), explain why. 35 min and NSF. Modified at SFU by S. Johnson and N Albeding, 2005, 2009, 2014.
7 Wokshop Physics II: Unit 21 Gavitational & Electical Potential Page 21-7 Autho: Piscilla Laws Gauss' Law fo Electical and Gavitational Foces Gauss' law states that the net electic flux though any closed suface is equal to the net chage inside the suface divided by ε o. Mathematically this is epesented by an integal of the dot poduct of the electic field and the nomal to each element of aea ove the closed suface: Φ = E da = qenc ε 0 The fact that electic field lines spead out so that thei density (and hence the stength of the electic field) deceases at the same ate that the aea of an enclosing suface inceases can ultimately be deived fom the 1/ 2 dependence of electical foce on distance. Thus, Gauss' law should also apply to gavitational foces. Activity 21-3: The Gavitational Gauss' Law State a gavitational vesion of Gauss' law in wods and then epesent it with an equation. Hint: If the electic field vecto, E, is defined as F e /q, let's define the gavitational field vecto Y as F g /m by analogy. We can use the new vesion of Gauss' law to calculate the gavitational field at some distance fom the suface of the eath just as we can use the electical Gauss' law to detemine the electic field at some distance fom a unifomly chaged sphee. This is useful in figuing out the familia foce "due to gavity" nea the suface of the eath and at othe locations.
8 Page 21-8 Wokshop Physics II Activity Guide SFU Activity 21-4: The Gavitational Foce of the Eath (a) Use Gauss' law to show that the magnitude of the gavitational field, Y, GM e at a height h above the suface of the eath is given by R e + h ( ) 2. Hints: How much mass is enclosed by a spheical shell of adius Re + h? Does Y have a constant magnitude eveywhee on the suface of the spheical shell? Why? If so, can you pull it out of the Gauss' law integal? (b) Calculate the gavitational field, Y, at the suface of the eath (h = 0). The adius of the eath is R e 6.38 x 10 3 km and its mass M e 5.98 x kg. Does the esult look familia? How is Y elated to the gavitational acceleation g? (c) Use the equation you deived in pat (a) to calculate the value of Y at the ceiling of the oom you ae now in. How does it diffe fom the value of Y at the floo? Can you measue the diffeence in the lab using the devices available? (d) Suppose you tavel halfway to the moon. What is the new value of Y? Can you measue the diffeence? (Recall that the eath-moon distance is about 384,000 km.) and NSF. Modified at SFU by S. Johnson and N Albeding, 2005, 2009, 2014.
9 Wokshop Physics II: Unit 21 Gavitational & Electical Potential Page 21-9 Autho: Piscilla Laws (e) Is the gavitational acceleation "constant", g, eally a constant? Explain. (f) In pat (d) you showed that thee is a significant gavitational attaction halfway between the eath and the moon. Why, then, do astonauts expeience weightlessness when they ae obiting a mee 120 km above the eath? HENRY! GRAVITATIONAL POTENTIAL Wok in a Gavitational Field A Review Let's eview some old definitions in pepaation fo tackling the idea of wok and enegy expended in moving though an electic field. Wok, like flux, is a scala athe than a vecto quantity. It is defined mathematically as a line integal ove a path between locations i and f. Each element of F d s epesents the dot poduct o pojection of F along d s at each place along a path.
10 Page Wokshop Physics II Activity Guide SFU W = i f F d s i F θ ds f F d s = F ds cosθ Let's eview the pocedues fo calculating wok and fo detemining that the wok done in a consevative foce field is independent of path by taking some wok measuements along two diffeent paths, ac and adc, as shown in the diagam below. In ode to take measuements you will need the following equipment: an inclined plane a cat a 5-N sping scale a mete stick a potacto Set the inclined plane to an angle of about 30 o to the hoizontal and take any necessay length measuements. Don't foget to list you units! Activity 21-5: Eathly Wok (a) Use path ac to aise the cat up an inclined plane that makes an angle of about 30 degees with the eath's suface (see diagam above). Use a sping and NSF. Modified at SFU by S. Johnson and N Albeding, 2005, 2009, 2014.
11 Wokshop Physics II: Unit 21 Gavitational & Electical Potential Page Autho: Piscilla Laws balance to measue the foce you ae exeting, and calculate the wok you pefomed using the definition of wok. (b) Raise the cat diectly to the same height as the top of the inclined plane along path ad and then move the cat hoizontally fom point d to point c. Again, use the definition of wok to calculate the wok done in aising the cat along path adc. (c) How does the wok measued in (a) compae to that measued in (b)? Is this what you expected? Why o why not? Hint: Is the foce field consevative? 15 min Wok and the Electic Field It takes wok to lift an object in the eath's gavitational field. Loweing the object eleases the enegy that was stoed as potential enegy when it was lifted. Last semeste, we applied the tem consevative to the gavitational foce because it "eleases" all of the stoed enegy. We found expeimentally that the wok equied to move a mass in the gavitational field was path independent. This is an impotant popety of any consevative foce. Given the mathematical similaity between the Coulomb foce and the gavitational foce, it should come as no supise that expeiments confim that an electic field is also consevative. This means that the wok needed to move a chage fom point A to point B is independent of the path taken between points. A chage could be moved diectly between two points o looped aound and the wok expended to take eithe path would be the same. Wok done by an electic field on a test chage q o taveling between points A and B is given by W = F d s = B q 0 E d A s A B
12 Page Wokshop Physics II Activity Guide SFU Activity 21-6: Wok Done on a Chage Taveling in a Unifom Electic Field (a) A chage q tavels a distance d fom point A to point B; the path is paallel to a unifom electic field of magnitude E. What is the wok done by the field on the chage? How does the fom of this equation compae to the wok done on a mass m tavelling a distance d in the almost unifom gavitational field nea the suface of the eath? B E d A (b) The chage q tavels a distance d fom point A to point B in a unifom electic field of magnitude E, but this time the path is pependicula to the field lines. What is the wok done by the field on the chage? E A d B and NSF. Modified at SFU by S. Johnson and N Albeding, 2005, 2009, 2014.
13 Wokshop Physics II: Unit 21 Gavitational & Electical Potential Page Autho: Piscilla Laws (c) The chage q tavels a distance d fom point A to point B in a unifom electic field of magnitude E. The path lies at a 45 angle to the field lines. What is the wok done by the field on the chage? A E d B
14 Page Wokshop Physics II Activity Guide SFU SESSION TWO: ELECTRIC POTENTIAL DIFFERENCE 15 min Potential Enegy and Potential Diffeence Recall that by definition the wok done by a consevative foce equals the negative of the change in potential enegy, so that the change in potential enegy of a chage moving fom point A to point B unde the influence of an electical foce can be found as follows: - U = W = F d s = q 0 E d s A B A B By analogy to the definition of the electic field, we ae inteested in defining the electic potential diffeence V = V B V A as the change in electical potential enegy U pe unit chage. Fomally, the potential diffeence is defined as the wok pe unit chage that an extenal agent must pefom to move a test chage fom A to B without changing its kinetic enegy The potential diffeence has units of joules pe coulomb. Since 1 J/C is defined as one volt, the potential diffeence is often efeed to as voltage. Activity 21-7: The Equation fo Potential Diffeence Wite the equation fo the potential diffeence as a function of E, d s, A, and B. 20 min The Potential Diffeence fo a Point Chage The simplest chage configuation that can be used to conside how voltage changes between two points in space is a single point chage. We will stat by consideing a single point chage and then move on to moe complicated configuations of chage. A point chage q poduces an electic field that moves out adially in all diections. The line integal equation fo the potential diffeence can be evaluated to find the potential diffeence between any two points in space A and B. and NSF. Modified at SFU by S. Johnson and N Albeding, 2005, 2009, 2014.
15 Wokshop Physics II: Unit 21 Gavitational & Electical Potential Page Autho: Piscilla Laws It is common to choose the efeence point fo the detemination of voltage to be at so that we ae detemining the wok pe unit chage that is equied to bing a test chage fom infinity to a cetain point in space. Let's choose a coodinate system so that the point chage is conveniently located at the oigin. In this case we will be inteested in the potential diffeence between infinity and some point which is a distance fom the point chage. Thus, we can wite the equation fo the potential diffeence, o voltage, as ΔV = V B V A = V V = E d s Often, when the efeence point fo the potential diffeence is at infinity, this diffeence is simply efeed to as "the potential" and the symbol V is just eplaced with the symbol V. Activity 21-8: Potential at a Distance fom a Chage Show that, if A is at infinity and B is a distance fom a point chage q, then the potential V is given by the expession V = kq y q 0 + Test chage being pushed in fom to a distance fom chage q. q Chage at oigin. x Hint: What is the mathematical expession fo an E-field fom a point chage?
16 Page Wokshop Physics II Activity Guide SFU 35 min The Potential Diffeence Due to Continuous Chage Distibutions The potential fom a continuous chage distibution can be calculated seveal ways. Each method should yield appoximately the same esult. Fist, we can use an integal method in which the potential dv fom each element of chage dq is integated mathematically to give a total potential at the location of inteest. Second, we can appoximate the value of the potential V by summing up seveal finite elements of chage q by using a compute speadsheet o hand calculations. Finally, we can use Gauss' law to find the electic field along with the defining equation fo potential diffeence to set up the appopiate line integal shown in Activity Again, let's conside a elatively simple chage distibution. In this case we will look at a ing with chage unifomly distibuted on it. We will calculate the potential on the axis passing though the cente of the ing as shown in the diagam below. (Late on you could find the potential diffeence fom a disk o a sheet of chage by consideing a collection of nested ings). A ing of chage has a total chage of Q = 20.0 µc. (i.e x 10-6 C). The adius of the ing, a, is 30.0 cm. What is the electic field, E, at a distance of x cm fom the ing along an axis that is pependicula to the ing and passes though its cente? What is the potential, V? Let's begin by calculating the potential. and NSF. Modified at SFU by S. Johnson and N Albeding, 2005, 2009, 2014.
17 Wokshop Physics II: Unit 21 Gavitational & Electical Potential Page Autho: Piscilla Laws Hints: Since the potential is a scala and not a vecto we can calculate the potential at point P (elative to ) fo each of the chage elements q and add them to each othe. This looks like a big deal but it is actually a tivial poblem because all the chage elements ae the same distance fom point P. Activity 21-9: Numeical Estimate of the Potential fom a Chaged Ring (a) Divide the ing into 20 elements of chage q and calculate the total V at a distance of x = 20 cm fom the cente of the ing. Show all of you wok below. Activity 21-10: Calculation of the Potential fom a Chaged Ring By following the steps below, you can use an integal to find a moe exact value of the potential. (a) Show that V = k dq = k dq x 2 + a 2
18 Page Wokshop Physics II Activity Guide SFU dq (b) Show that k = x 2 + a 2 x 2 + a 2 k x 2 + a 2 dq (i.e. show that is a constant and can thus be pulled out of the integal). (c) Pefom the integation in pat (b) above. Then substitute values fo a, x, and Q into the esulting expession in ode to obtain a moe "exact" value fo the potential. (d) How does the "numeical" value that you obtained in Activity 21-9 compae with the "exact" value you obtained in (c)? and NSF. Modified at SFU by S. Johnson and N Albeding, 2005, 2009, 2014.
19 Wokshop Physics II: Unit 21 Gavitational & Electical Potential Page Autho: Piscilla Laws Now let's take a completely diffeent appoach to this poblem. If we can find the vecto equation fo the electic field at point P due to the ing of chage, then we can use the expession ΔV = V (x) V = x E d s as an altenative way to find a geneal equation fo the potential at point P. de Activity 21-11: V fom a Ring using the E-field Method (a) Show that the electic field at point P fom the chaged ing is given by E = kqx ( x 2 + a 2 ) 3 2 î = kqx x 2 + a 2 3 î Hints: (1) Thee is no y component of the E-field on the x-axis. Why? x (2) cosθ = x 2 + a 2
20 Page Wokshop Physics II Activity Guide SFU (b) Use the equation ΔV = V (x) V = x E d s to find V. (c) How does the esult compae to that obtained in Activity (c)? 10 min Equipotential Sufaces Sometimes it is possible to move along a suface without doing any wok. Thus, it is possible to emain at the same potential enegy anywhee along such a suface. If an electic chage can tavel along a suface without doing any wok, the suface is called an equipotential suface. Conside the thee diffeent chage configuations shown below. Whee ae the equipotential sufaces? What shapes to they have? Hint: If you have any compute simulations available to you fo dawing equipotential lines associated with electical chages, you may want to check you guesses against the pattens dawn in one o moe of the simulations. and NSF. Modified at SFU by S. Johnson and N Albeding, 2005, 2009, 2014.
21 Wokshop Physics II: Unit 21 Gavitational & Electical Potential Page Autho: Piscilla Laws Activity 21-12: Sketches of Electic Field Lines and Equipotentials (a) Suppose that you ae a test chage and you stat moving at some distance fom the chage below (such as 4 cm). What path could you move along without doing any wok, i.e. E d s is always zeo? What is the shape of the equipotential suface? Remembe that in geneal you can move in thee dimensions. (b) Find some equipotential sufaces fo the chage configuation shown below, which consists of two chaged metal plates placed paallel to each othe. What is the shape of the equipotential sufaces?
22 Page Wokshop Physics II Activity Guide SFU (c) Find some equipotential sufaces fo the electic dipole chage configuation shown below. - + (d) In geneal, what is the elationship between the diection of the equipotential lines you have dawn (epesenting that pat of the equipotential suface that lies in the plane of the pape) and the diection of the electic field lines? 30 min Expeiment on Equipotential Plotting The pupose of this obsevation is to exploe the patten of potential diffeences in the space between conducting equipotential sufaces. This patten of potential diffeences can be elated to the electic field caused by the chages that lie on the conducting sufaces. You will be using pieces of cabonized pape with conductos painted on them in diffeent shapes to simulate the patten of equipotential lines associated with metal electodes in ai. (Waning: this is only a simulation of "eality"!) One of the papes has two small cicula conductos painted on it and the othe has two linea conductos. and NSF. Modified at SFU by S. Johnson and N Albeding, 2005, 2009, 2014.
23 Wokshop Physics II: Unit 21 Gavitational & Electical Potential Page Autho: Piscilla Laws Cicula electodes Line electodes Conducting Paint Cabonized Pape Refeence points fo the potential. Place the negative black pobe at the indicated points duing all measuements. You can use a battey to set up a voltage (i.e. a potential diffeence) acoss a pape with two electodes painted on it. You can then use a digital voltmete to tace seveal equipotential lines on it. To do this obsevation you will need the following equipment: equipotential plotting pape with line and cicula electodes foam boad metal push pins a digital voltmete 4 D-cell batteies 2 alligato clip leads Pick a piece of cabonized pape with eithe the two cicula electodes o the line electodes. Use the alligato clips to connect the teminals of the battey to each of the electodes on the pape as shown below.
24 Page Wokshop Physics II Activity Guide SFU Refeence electode Black Battey Batt Red Digital Voltmete Alligato clip Tun on the digital voltmete. Set the tip of the black voltmete pobe (plugged into the COM input) on the cente of the negative cicula electode. Place the ed pobe (plugged into the V-Ω-S input) on any location on the pape. The eading on the voltmete will detemine the potential diffeence between the two points. What happens when you evese the pobes? Why? Afte placing the black pobe on the cente of an electode, you will use the ed pobe to find the equipotential lines fo potential diffeences of +1V, +2 V, +3 V, +4V and +5 V. Activity 21-13: Mapping the Equipotentials (a) On the two sheets of gid pape povided daw diagams of the two electode sheets you ae going to examine. (b) Fo both electode sheets, sketch the lines you mapped to scale on the diagams. Attach these sheets to this Activity Guide when you hand it in. (c) Compae the esults to those you pedicted in Activity Is it what you expected? If not, explain how and why it diffes. Make sketches in the space below, if that is helpful. and NSF. Modified at SFU by S. Johnson and N Albeding, 2005, 2009, 2014.
25 Wokshop Physics II: Unit 21 Gavitational & Electical Potential Page Autho: Piscilla Laws NOTE: BE SURE TO TURN THE VOLTMETER OFF AND DISCONNECT THE BATTERIES WHEN FINISHED WITH OBSERVATIONS.