Notes follow and parts taken from sources in Bibliography

Transcription

1 PHYS 33 Notes follow and pats taken fom souces in Bibliogaph ectos The stud of electomagnetism equies the abilit to manipulate vecto quantities, since both electic and magnetic fields ae vecto fields. Besides the addition and subtaction of vectos that ou have used since Phsics I, moe complicated opeations ae also possible. One impotant thing to keep in mind is the distinction between vectos and scalas. While a scala is just a single numbe, a vecto (in 3D) is a collection of thee numbes. This does not mean that a vecto can be consideed to be thee scalas witten togethe. The eason is that scalas ae invaiant (meaning the do not change) unde otations. ecto components ae not invaiant unde these otations. One of the simplest eamples of this was demonstated when ou fist encounteed the block on an inclined plane in intoducto phsics. You had the choice of oienting ou coodinate sstem so that & wee hoizontal and vetical, o choosing them so that the wee paallel and pependicula to the inclined plane. The and values of the block s displacement while sliding down the plane ae diffeent in the two coodinate sstems. Theefoe, the displacement vecto has diffeent components in each coodinate sstem. A pai of scalas, on the othe hand, would be the same in both coodinate sstems. If ou chose to wite the block s mass and cost as a pai (5.5 kg, $.38), that pai is obviousl independent of ou choice of coodinates. Although the vecto components ae coodinate-dependent, we can fom an invaiant object fom the vecto b calculating its length. The vecto opeation hee is the dot poduct of the vecto with itself. Since the definition of the dot poduct of vectos A and B in D Catesian coodinates is A B A B The length of the vecto is just the squae oot of the dot poduct, and it is invaiant. In fact, the dot poduct between an two vectos is an invaiant, as we will show. Call the two coodinate sstems (,) and (, ). A B (,) (, ) The length of the plank is L in each sstem. Obviousl, the path taken in the second coodinate sstem is just (,L). In the fist coodinate sstem, though, the block would stat at, L Sin(θ) and finish at L Cos(θ),. B the Pthagoean theoem, the distance taveled is still L. As a geneal ule, an vecto in the fist coodinate sstem could be tansfomed to a vecto in the second sstem if we use the fomula below:

2 PHYS 33 ( A A ) ( A ' A ') Cosθ Sinθ Sinθ Cosθ This tells us that we can wite the components of the vecto in the second coodinate sstem as A ' ACosθ ASinθ and A ' ASinθ A Cosθ The dot poduct in the unpimed fame, which is A B A B b definition, would be A B A B (again b definition) in the pimed fame. Ae the equal? Do the same tansfomation as above on the B vecto and then calculate A B A B : ( A Cosθ A Sinθ )( B Cosθ B Sinθ ) ( A Sinθ A Cosθ )( B Sinθ B Cosθ ) If ou do the math coectl, ou ll be left with A B Cos θ A B Sin θ A B Cos θ A B Sin θ A B A B just as ou got in the unpimed fame b definition. It s impotant to note that this paticula fom of the dot poduct holds fo Catesian coodinates onl. As ou can see fom these tansfomations, the dot poduct is a wa to tun two vectos into a single scala. The mati hee is a otation mati, and it has an inteesting popet. If we multipl the tanspose of a otation mati b that otation mati, we get Cosθ Sinθ Sinθ Cosθ Cosθ Sinθ Sinθ Cosθ This is because the invese of the otation mati is the same as the tanspose of the otation mati, which we could wite as R - R T. The invese of the otation mati undoes the otation (b definition) and we can see that it s the same as changing the angle fom θ to -θ. Two vectos can also be multiplied to fom a thid vecto (technicall, a pseudovecto). This is the coss poduct, and in intoducto phsics, we just epesent it as the poduct of the lengths of the two vectos and the sine of the angle between them. We can also wite it in component fom, and the easiest wa to do that is to epesent it as the deteminant of a 3 3 mati shown below:

3 PHYS 33 ˆ A B ˆ A B zˆ A B z z A B This can be witten out to get (A B z A z B ) (A z B A B z ) (A B A B )z. Geometicall, ou can conside the coss poduct to be the aea of the plane bounded b vectos A and B, just as the dot poduct is equivalent to the pojection of A on B (o B on A). Combining these ideas, we can find the volume bounded b thee vectos A, B, and C, A B C. using the scala tiple poduct ( ) It will also occasionall be useful to find the vecto tiple poduct of A, B, and C, which is A B C. We can ewite this, if necessa, as just ( ) A ( B C ) B( A C ) C ( A B) Cleal, the esult of this opeation will be a fouth vecto. In all cases, the ke facto that detemines whethe an object is a vecto o not is the object s behavio unde otations/ tansfomations. The otation mati Cosθ Sinθ Sinθ Cosθ is applied to the components of a vecto as shown ealie to find the components in the new (otated) fame. In thee dimensions, a vecto will have thee components and the otation mati will be a 3 3 mati. We can conside a vecto to be a special case of a class of objects known as tensos. A vecto is a tenso of ank one, a scala is a tenso of ank zeo, and we can also have tensos of highe anks. Fo eample, an object s moment of inetia depends on the ais about which ou t to otate it. That s wh, when ou leaned that the moment of inetia of a disk was ½ MR in Phsics I, the diagam also specified that this was fo otation aound the ais of smmet. Rotating it aound a diffeent ais would give a diffeent value of moment of inetia. Fo that eason, the moment of inetia is eall a tenso of ank two, meaning ou need a 3 3 mati to descibe it full. Note that although the might be witten on a -D piece of pape, we could have tensos of ank 3, 4, etc. In geneal, a ank-n tenso will have 3 N components in 3-D and tansfoming it to a new coodinate sstem will equie multiplication b N 3 3 otation matices. 3

4 PHYS 33 Diffeential Calculus In one dimensional calculus, the deivative is simple. It epesents the change in one vaiable as anothe is changed, and can theefoe be viewed as the slope of a line plotting the fist vaiable as a function of the second. If we look at quantities in thee dimensions of space, the idea of the deivative gets moe complicated. Assume ou have a scala function of position (tempeatue is the eample used in ou book and is the most common one) and want to know how it changes as ou move fom point to point in the oom. The total change in tempeatue can be witten as a sum of thee diffeent changes. We can wite this as dt T d T d T z dz whee now we use patial deivatives to indicate that T is eall a function of all thee vaiables,, and z but we want to hold and z constant while diffeentiating with espect to, etc. If ou look at the fom of this equation, ou can see that it kind of looks like a dot poduct. One pat would be a diffeential length dl (d, d, dz) and the othe would be the action of the gadient opeato on the tempeatue field. Even though tempeatue itself is a scala, the gadient acting on it will poduce a vecto. We could wite it as T T ˆ T ˆ T z zˆ Geometicall, we can conside this T to point in the diection of maimum incease of T and the magnitude of T gives a value fo the slope. Of couse, unless T is an etemel simple tempeatue field, T will have diffeent values at diffeent points in the oom. Note that both the diection and magnitude of T can change. Although is not itself a vecto, we can teat it as one fo a vaiet of uses. The most geneal wa to wite it is then ˆ ˆ zˆ z Witing it this wa makes it clea that it is an opeato and doesn t eall make an sense until it is applied to a function. Also, note that the unit vectos ae witten to the left of the patial deivative opeatos, since we don t want to appl that opeation to the unit vectos themselves. 4

5 PHYS 33 If we can teat this thing like a vecto, we can ask what we would get if we applied it to anothe vecto in a dot poduct. We would wite it like A A A A z z and this object is called the divegence of A. This is connected with the idea of souces o sinks of a vecto field. If we wee consideing the electic field, fo eample, souces of it would be positive chages and sinks of it would be negative chages, since E field lines flow out of positive chages and into negative chages. A unifom vecto field, fo eample, would have zeo divegence since it s the same evewhee and theefoe does not flow into o out of an point. If we again teat the gadient opeato like a vecto and find the coss poduct of it with some vecto A, it would look like this: A ˆ A ˆ A zˆ z A z and we efe to this as the cul of A. This is because it epesents the degee to which the vecto A culs aound the point whee this quantit is evaluated. Because the magnetic field B culs aound a cuent-caing wie, we ll see late that we can wite B as being popotional to the cuent in the wie. You book (o an book at this level on electomagnetism) has a vaiet of identities elating to this gadient. You ll find them to be useful when doing homewok poblems. Thee ae seveal was we might see this gadient opeato applied twice. If we take the divegence of the gadient, we get ( ) T T T T z This opeation is also known as the Laplacian and we would usuall wite it (opeating on the scala T) as T. If applied to a vecto, it would be applied to each component individuall (as if each component wee a scala, which we know is wong). If we use the coss poduct instead, we find that ( T ) 5

6 PHYS 33 Othe possibilities ae the gadient of the divegence ( ( T)), the divegence of the cul ( ( A), which is also alwas zeo), and the cul of the cul, which gives These identities will also eappea late. ( A) ( A) A Integal Calculus The integals we will use can be divided into those ove lines, those ove sufaces, and those ove volumes. In each case the measue will be diffeent usuall we use something like dl fo lines, da o ds fo aeas, and d o dτ fo volumes. Note that the line and the aea ae witten as vectos while the volume is a scala. We should have epected this since ou ealie wok with the coss poduct gave us a vecto epesenting an aea and the tiple poduct (A (B C)) was a scala. A line integal involving a vecto is geneall of the fom b a v dl ove the path fom a to b, but if the endpoints ae the same (meaning the path is closed) it will be witten as v dl whee the loop indicates that the path is closed. A suface integal would be moe like S v da whee S is the suface to be integated ove. Again, it could also be a closed suface: The simple volume integal would just be v da 6

7 PHYS 33 T d ol The fom of the measue is diffeent fo line, suface, o volume integals, but it is also diffeent fo diffeent coodinate sstems. As ou can again find in the font cove of ou book, the line elements and volume elements ae Coodinate sstem Line element olume Element Catesian d i d j dz k d d dz Spheical d dθ θ sinθ dφ φ sin θ d dθ dφ Clindical d dφ φ dz z d dφ dz Notice that thee is no suface element in this list. The eason is that the suface could be oiented in a vaiet of was, so ou ll have to detemine this on a case-b-case basis. Geneal esults of integal calculus that will be useful ae detailed below. Fist, we have b a ( T ) dl T ( b ) T ( a ) Note that the esult on the ight is independent of the paticula path taken, and onl depends on the actual endpoints of the path. Anothe impotant theoem (usuall called Gauss s theoem) is ol ( v ) dτ Suf v da This is a ve significant theoem, since it lets ou change the integal on the left (which depends on the entie volume in question) to an integal ove onl the suface which bounds that volume. This is the souce of what was efeed to as Gauss s law in intoducto electostatics as well as the simila statement in Newtonian gavitation. It tells us that the flow of something into o out of a volume is equivalent to finding the amount of that thing cossing the bounda of the volume. Related to this is Stokes s theoem. Hee we wite Suf ( v ) ds Path v dl Notice that we can use the same path to bound a vaiet of sufaces, just as the same suface can bound a vaiet of volumes. The tpical eplanation of this (as shown in ou 7

8 PHYS 33 book) is to imagine the suface (ds) as containing man ciculating loops (fom the v tem). In the inteio, each of those loops will be canceled b its neaest neighbos, as ou can see below: As this will happen at all points, the onl suviving pats will be at the bounda of the suface, whee thee is not anothe neighbo to cancel it. Fo this eason, the sum of all the ciculating loops thoughout the suface educes to a line integal aound the bounda. Coodinate Sstems While Catesian coodinates ae pobabl the most familia to eveone, othe coodinate sstems ae useful as well, especiall when the poblem (i.e., the chage o cuent distibution) has the same smmet as the altenate coodinate sstem. Spheical pola coodinates ae one of the possible thee-dimensional etensions of odina plana pola coodinates. Hee, instead of,, and z, ou vaiables ae, θ, and φ whee is the distance to a point fom the cente of the coodinate sstem, θ (which is kind of like latitude on a globe) is measued fom the Noth pole, and φ (like longitude) is measued aound the equato. Keep in mind that thee ae othe conventions fo establishing this coodinate sstem whee φ and θ ae swapped. See the image below fo moe details. Z θ φ X Y 8

9 PHYS 33 The blue point would have Catesian coodinates (,,z) and spheical coodinates (,θ, φ). What is the tansfomation back and foth between them? Looking at the geomet, we could find z Sinθ Cosφ θ φ Cos Tan z and Sinθ Sinφ z Cosθ Knowing this also lets us constuct a mati elating the two coodinate sstems, known as the Jacobian. This tells us how the coodinates of sstem A change with espect to those of sstem B. In othe wods, it s just a mati of all patial deivatives between the two coodinate sstems. It can be calculated quickl and shown to be J z θ θ z θ φ φ z φ Sinθ Cosφ Sinθ Sinφ Cosθ Cosθ Cosφ Cosθ Sinφ Sinθ Sinθ Sinφ Sinθ Cosφ Multipling this mati with its tanspose (the mati witten with ows and columns swapped) gives us a useful mati known as the metic (usuall epesented b g), which is what detemines the fom of the line element, suface element, volume element, etc. The eason ou ma neve have seen the metic befoe is that, fo Catesian coodinates, it s just the identit mati. It was thee, but ou wouldn t notice it. Fo this case, we get J T J g Sin θ To find the volume element, we would take the squae oot of the deteminant of this mati and then multipl b the diffeential of each coodinate. Since ou deteminant hee is 4 Sin θ, the volume element should just be 4 Sin θ d dθ dφ Sin θ d dθ dφ 9

10 PHYS 33 The line element is found b assigning the fist ow/column to d, the second to dθ, and the thid to dφ. We can then ead it off as the sum of d dθ Sin θ dφ. Note that this same pocess applied to Catesian coodinates gives the familia volume element to be d d dz and the line element to be d d dz. Clindical coodinates ae just pola coodinates in the plane with the z ais being the same in both sstems. Again, it will have a diffeent metic, diffeent line element, and diffeent volume element. These epessions ae also in the font of ou book, along with the clindical coodinate epesentations of the cul, divegence, gadient, and Laplacian. Diac Delta Function Because we get so much use out of the concept of chages as point chages, we need a wa to descibe them mathematicall. As a side note, the idea of a point chage is not at all caz; accoding to all available evidence, the electon has no phsical etent; it eall is a point chage. The poton is of couse a composite paticle, but at enegies that ae small on the scale of the poton s est mass, we can conside it to be a point chage as well. Thee ae some poblems with this appoach; if we t to calculate the electic field as we move close in towads the chage, it will incease without limit. We will get infinite values fo both chage densit and electic field. Thee will have to be a wa to epesent this that still gives phsicall easonable esults that match epeiment. If we look at the geneal fom of the electic field of a point chage (ignoing the constants and the size of the chage), it would look like ˆ Calculating the divegence of this function (and spheical coodinates would be the obvious choice to take advantage of the spheical smmet of this poblem) would give which is cleal zeo. Howeve, if we take the volume integal of this quantit (divegence of the vecto field), it could also be eplaced (using Gauss law) b the suface integal of the vecto field ove a closed suface encompassing that point. Fo a spheical suface centeed on that point, we get v da ˆ ( R sin θ dθ dφ ˆ ) 4π R

11 PHYS 33 Notice that this is not equal to zeo. To fi this disageement, we need to intoduce a special kind of mathematical device known as the Diac delta function (which is actuall a distibution of functions, but that s not paticulal impotant fo us). This is defined as δ ( ) if if A moe caeful definition would be that we have some kind of smmetic function (ectangle, tiangle, etc.) scaled such that its aea. We then make it both skinnie and talle (while keeping the aea constant). In the limit that its width, the height has to tend to. That s the delta function. This function is eall onl useful when it s inside an integal. When the limits of integation include zeo, the delta function picks out the value at zeo of whateve else is inside the integal. Fo eample: If ou instead want to pick out the value of f at some othe point on the ais, ou can shift the location of the delta function b ewiting it as δ(-a). The point of inteest will still be the place whee the agument is zeo, but that s now at a. When ou think about a phsical analog of this function, it would be ou point chage. If ou integate ove a suface bounding a volume that contains this point chage, ou see that it is thee. If ou look anwhee othe than the oigin, ou would see zeo, and at the oigin, an infinite concentation of chage. The thee-dimensional vesion of this can be constucted b just multipling thee -D functions: δ 3 ( ) δ ( ) d f ( ) f ( ) δ ( ) δ ( ) δ ( z ) The integal of this function in a volume that includes the oigin gives a value of. With this new idea in place, we can go back and fi the poblem at the stat of this section. We find now that the divegence we calculated ealie should eall be witten as ˆ 4 π δ 3 ( ) Potentials Fo cetain kinds of fields, we can take advantage of vecto identities to define elated quantities called potentials. Fist, if we have a field with a cul of zeo, it can alwas be witten as the gadient of a scala function. The pima eample of this is found in electostatics, whee we can wite

12 PHYS 33 E meaning we can define a potential (the electostatic potential, also known as the voltage) and wite E as the gadient of that potential (though b convention we wite it as the negative of the gadient of the potential): E The advantage to this is that we can find the electic field fom a complicated assotment of chages much moe easil using the potential and its gadient than would be the case if we tied to do it with vecto addition of each electic field. Othe implications of a cul-fee field (also called iotational) include the path independence of E dl and the fact that the integal of the same quantit aound a closed loop is zeo. On the othe hand, if the divegence of a vecto field is zeo, it can be witten as the cul of anothe vecto field (called the vecto potential). The most impotant eample of this theoem is found in the magnetic field. Because thee ae no magnetic chages, the divegence of the magnetic field is zeo. We can theefoe wite it as the cul of the magnetic vecto potential, usuall epesented as A. In othe wods, B A. In this case, we can sa that the integal of B da ove an closed suface is zeo. In each of these cases, the potential is not completel specified b what we have said so fa. Fo the electostatic potential, we could add an constant value to it without changing the value of the E field since the deivative of the scala function is not changed b adding a constant to that function. This is the same idea as the concept that the zeo of gavitational potential eneg can be set wheeve ou want without changing the phsical situation. Fo the case of the magnetic field, an quantit with zeo cul could be added to the vecto potential without changing the B field. As we just saw, the gadient of a scala function has no cul. Fo that eason, the vecto potential A can have the gadient of a scala function (like the electostatic potential, fo that matte!) added to it without changing the value of B. In geneal, an vecto field can be witten as a combination of the cul of anothe vecto field plus the gadient of a scala field. This is simila to the wa that a geneal 3-D vecto could be witten as a pat paallel to the ais and a pat pependicula to that ais. If ou vecto potential is and ou scala field is S, we can wite an vecto field F as F S

13 PHYS 33 Electostatics The logical stating point fo studing electicit and magnetism is pobabl electostatics. It has quite a bit in common with mateial alead coveed in mechanics couses. If the chages ae at est (static) we onl have electic fields. Since we haven t eall intoduced them et, though, we can back up a step and stat with Coulomb s law. This is the electostatic equivalent of Newton s law of gavit, since it descibes the foce of attaction (o, unlike the gavitational case, epulsion) between two chages. We can wite this as F 4π qq ˆ Notice that the foce is along the line joining the two chages and dops off as the invese squae of thei sepaation. As ou saw in, the constant is known as the pemittivit of fee space and has a value of C /(N m ). The othe ke concept we ll need is the pinciple of supeposition. We ve used this etensivel in pevious couses; all it sas is that the foce between two chages is not changed b the addition of a thid. Of couse, the total foce on each of the chages will change when the thid is intoduced, but onl because that new chage is itself eeting a foce on each of the othe two. This is eall the simplest possible situation, so it s fotunate fo us that that s how natue woks. Just as the concept of a gavitational field is useful in mechanics, the concept of an electic field is useful in electostatics. Essentiall, it lets us calculate the foce (magnitude and diection) that would be felt b a hpothetical test chage (geneall chosen to be positive and infinitesimal, both in phsical size and amount of chage contained) at an given point. If a nc chage would feel a foce of µn to the West at a cetain point, we can sa that the electic field points to the west and has a size of ( µn/ nc) N/C. This is a little moe convenient than descibing things in tems of the foces themselves, since that would equie us to have a standad test chage that eveone can agee to use. This wa, the size of the test chage dops out and that s one less thing to wo about. Using this definition, we get that the electic field can be witten as E F / q Similal, the electic field poduced b an assotment of N chages will just be the vecto sum of the N foces eeted on a test chage q divided b the size of q. We could wite the vecto field E at point P as E 4π N ( P) i q ˆ i i i 3

14 PHYS 33 If the chages ae not discete, but continuous, we move fom the sum to an integal. You might object to this, since we leaned in that thee is indeed a smallest unit of electic chage (eithe the chage on the electon, o one thid of that, depending on ou point of view) and theefoe thee is no such thing as a continuous distibution. That would be coect, but we will see that a discete sum of a huge numbe of tin chages is effectivel indistinguishable fom an integal, and the integal is much easie to use. Tanslating the above fomula, we would get ˆ dq 4 π ( P) E meaning we e doing the same thing as befoe we e looking at each tin little element of chage and finding the electic field it causes at some point P, and finding the vecto sum of all of those effects. As ou book shows, the dq pat can take diffeent foms. If ou e looking at a line chage, ou ll pobabl see it as dq λ dl whee dl is now an infinitesimal length of that line and λ is the chage densit (usuall in C/m). Fo a suface chage, it would be dq σ da, whee da is the suface element and σ is the suface chage densit (C/m ). Finall, we could also wite dq ρ d o ρ dτ fo a volume chage distibution. The paticula fom of dl, da, o d (o dτ) depends on ou coodinate sstem (and the oientation of the suface, in the σ da case). Fields and Potentials The field lines due to point chages ae adial (adiall outwad fo positive chages and adiall inwad fo negative chages). Field stength is epesented b the densit of the field lines, which will dop off as / as ou move awa fom the chage. This is eas to see if ou imagine a fied numbe of lines N leaving a positive chage, fo eample. If a sphee of adius suounds that positive chage, the line densit will have to be N/(sphee aea) o N/(4 π ). The value of N won t change as ou move awa, but the aea of the sphee inceases as, so the aeal densit of field lines (and theefoe the field stength) deceases as /. Note that -D epesentations of this would onl show a / dopoff in line densit since it would then be numbe of lines/ peimete of a cicle. The field lines can t coss, since the epesent the magnitude and diection of the acceleation that would be felt b a positive chage placed at a point. If the lines cossed, the chage wouldn t know which wa to go. A elated and impotant concept is the flu. This is a measue of the component of the E field pependicula to a given suface. We can wite it mathematicall as Φ E S E da 4

15 PHYS 33 You can think of this as being like the foce on a sheet of plwood on a wind da. The foce will incease if the wind speed is highe (E is lage), if the plwood is lage (integal of da taken ove a lage suface), o if the wood is pependicula to the wind (meaning the angle between the nomal to the boad and the wind s diection is º). If the suface is closed (like the suface of a sphee), the flu into o out of it depends onl on the chage within that closed suface. This is the essence of Gauss law, which we can wite as S E da Q enc The impotant thing hee is that we can find out about an net chage in the inteio of the suface just b making measuements on (o outside of) the suface. This is a fail emakable esult. We could wite a simila fomula fo gavit, and one of the implications of it is that, if we want to pedict ou weight fom ou mass, we don t have to calculate the foce vectos between eve atom of ou and eve atom of the Eath and then sum them up. We can instead teat the Eath s mass as if it wee a point mass located at the cente of the Eath and look at the attaction between it and ou with a single application of Newton s law of gavit. This fomula makes a simila eduction in wok possible in electostatics. Rathe than doing a long and involved integal to find E, we can find it using Gauss law if the poblem has sufficient smmet. If we use the divegence theoem on this esult, we can ewite it as S E da ( E ) Howeve, since we could wite the enclosed chage Q enc as an integal of chage densit ove the same volume, ρ dτ ( E ) dτ dτ we could take awa the integals and we would then get E ρ 5

16 PHYS 33 This is the diffeential fom of Gauss law, which is sometimes moe useful than its integal fom (the fom ou saw in PHYS ). The integal fom would pobabl be moe useful when ting to find the E field aound a smmetic chage distibution. Fo eample, if ou have a long chaged clinde of adius R with a chage densit of the fom ρ A 3 whee A is some constant, ou can find the E field eithe inside o outside of the distibution using Gauss law. Inside, the field at a distance fom the ais is found b constucting an imagina clinde (a Gaussian suface) that is coaial with the phsical clinde and has a adius equal to. Gauss law then gives us E da S A 3 d The aea element fo the suface of a clinde will have no d tem since the adius of ou Gaussian suface is constant. We will measue the aea in the θ and z diections, so the aea element will be dθ dz. On the ight, we need the full volume element fo a clinde, which we found peviousl to be d dθ dz. The potentiall complicated pat of this pocess is the dot poduct inside the integal on the ight. Fotunatel, we can use the smmet of the poblem to make this simple. Because the clinde is smmetic, thee can t be a special point anwhee on it; theefoe, the E field has to have the same magnitude evewhee on the suface of the Gaussian clinde and it will also have to be adiall diected (adiall out if positivel chaged, adiall in if negative). This means that we can take the E outside of the integal on the left, and we can eithe ignoe the dot poduct (because cos ) o eplace it b a negative sign (cos 8º -), depending on the chage. We ll assume that the chage is positive. This leaves us with the (much simple) fomula below: E A ( ' ) ' d' dθ ' dθ dz dz 3 Notice that the integals on the left side ae just the suface aea of a clinde; when we calculate them, we ll get π L fo a clinde of length L. On the ight, because the chage distibution is independent of both θ and z, we can calculate those two integals to be π L. All that s left is the integal ove, which we quickl find to be 5 /5. The net esult is then π L E A π 5 5 L 6

17 PHYS 33 o E A 5 4 inside the clinde. If we look outside instead, we follow the eact same pocedue with the same aguments. The left side (integal ove Gaussian suface) is the same, but the integal on the ight is diffeent because the chage distibution no longe fills the Gaussian suface. It will stop at a adius of R athe than continuing on out to (emembe that >R outside the phsical clinde). That means the cancellation of on each side won t happen, and we will instead get π L E A π R 5 5 L educing to E A 5 R 5 Notice that eithe epession gives the same E field at the R bounda. This will alwas be the case unless a suface chage is pesent. We ll see that this happens with conductos. Could this clinde be a conducto? Wh o wh not? As in the book, we can calculate the cul of the E field of a point chage (indiectl, b finding the integal of E aound a closed path). Because E doesn t depend on eithe θ o φ (smmet again), the onl pat of the dot poduct that mattes is the d tem. We can then wite b a q q d 4π 4π Of couse, this will be zeo fo a closed path whee ab. It is equall clea that E if ou look at the fomula fo the cul in spheical coodinates in the font cove of the book. The onl component of E is the adial component, and the cul onl involves deivatives of E taken with espect to θ and φ, so it is tiviall zeo. a b 7

18 PHYS 33 The Electic Potential As we saw when eviewing vecto calculus, if a vecto field is iotational (has zeo cul), it can be epessed as the gadient of a scala field. When that vecto field is the electic field, the scala field is known as the electic potential. We need a efeence point fo ou measuement of electic potential. B compaison, we can look at the gavitational case. If we emain nea the suface of the Eath, we can wite the gavitational potential (which ael pops up in intoducto phsics) as gh. The height has to be measued elative to something sea level, o the floo, o whateve. The value of the height itself has no meaning; the onl impotant thing is the diffeence in height between two places. Similal, we have to pick some efeence point fo the electic potential, and it is customa to pick infinit as the zeo of potential. This makes sense, as the influence of the chage distibution should be zeo if ou ae infinitel fa awa fom it. The epession fo the potential is then ( ) E ef dl whee ef is the efeence point. We could equivalentl wite E One of the big advantages to using the potential to descibe a poblem is the fact that it is a scala. We can then find the electic field calculating the (negative) gadient of this potential. Incidentall, although the gavitational potential ael appeas in intoducto phsics, it is a ke element of geneal elativit. The change in the ate of the flow of time in a gavitational field depends on the gavitational potential athe than the gavitational field. If we combine this esult fo potential with the ealie esults fo the divegence of E and vecto calculus, we can wite ρ (Poisson s equation) and, when thee is no chage pesent,, a special case known as Laplace s equation. Note that we can see the eason fo the negative sign in Poisson s equation and ou connection between and E easil hee: a positive point chage would mean the quantit on the ight (-ρ/ ) is negative. If the second deivative of a function is negative, the function is a maimum thee. In othe wods, the electic potential is a maimum at a positive chage and a minimum at a negative chage. 8

19 PHYS 33 Calculating the potential fom the chage distibution equies eithe a sum (point chages) o an integal (chage distibution). Fo a single point chage, we get ( ) q 4π When thee ae multiple point chages, because of the pinciple of linea supeposition, we just collect tems like the one above and add them (eas, since the e scalas). In the continuous case, the basic idea would be 4π ( ) dq whee dq can be eplaced b λ dl fo a line chage, σ da fo a suface chage, o ρ d fo a volume chage distibution. Bounda Conditions The ules we have alead seen place some estictions on the behavio of the electic field and electic potential acoss the boundaies between mateials. Analzing the electic field is made easie b the use of Gauss law. If we have a continuous distibution of chage, such as the clinde with a chage distibution of the fom A 4 as befoe, we saw that both fomulas agee at the bounda. This means the electic field is continuous at the bounda in that eample. If we instead conside a sheet with a suface chage distibution σ, defined as a total chage Q divided b the sheet s aea A, we will see a diffeent esult. Ou Gaussian suface (shown in puple below) can be chosen to be a ectangula bo with two sides paallel to the sheet and the othes pependicula to it: In this case, as we shink the bo thickness down, we still have a chage contained in it. We can make the bo abitail thin and still have a chage inside, meaning thee will be a diffeence between the left and ight sides of the sheet. Theefoe, the E field is discontinuous at the bounda. Again using smmet, the E field has to be pependicula 9

20 PHYS 33 to the chaged sheet. This means thee will be no contibution to the flu on the fou sides of the bo pependicula to the sheet. Putting this togethe, we get that E above E below σ / In othe wods, the pependicula component of the E field is continuous unless thee is a suface chage pesent. If we look at the component of E paallel to the suface (also called the tangential component), we can take advantage of the fact that E is iotational and find the path integal fo a path that goes just above the suface in question, dips down into it, and etuns to its oigin. The pieces of the path pependicula to the suface contibute nothing to the path integal of E (again b smmet). Because the two emaining pats of the path ae tavesed in opposite diections, the will contibute with opposite signs, leaving us with E above L Ebelow L We quickl see that the tangential component of E has to be the same on each side of the bounda, meaning it is continuous. As ou book illustates, we can find out what happens to the potential b looking at the line integal of the E field fo a shot path that just cosses the suface. Because it can be made as shot as ou like, the integal can be made to appoach zeo. Theefoe, the potential is continuous acoss the suface. Wok and Eneg It s impotant to undestand the distinction between the electic potential and the electic field. If the field is zeo, that means thee is no electic foce on a chage at that point. If the potential is zeo, that means that a chage could be moved in fom infinit (whee the potential is defined to be zeo) without epending (o havesting) an eneg. The fact that the field is zeo at some point tells ou nothing, howeve, about the eneg it would take to move a chage to that point fom infinit. Likewise, a chage a point whee the potential is zeo could epeience a foce in an diection o no foce at all. As the electic field can be viewed as the foce pe unit chage, the electic potential can be seen as the potential eneg pe unit chage. The wok done to move a chage fom one place to anothe can be witten as W ( ) Q final initial If the initial position is infinit, and we define the potential to be zeo thee, we can just wite

21 PHYS 33 W Q To find the wok done assembling a goup of point chages, ou can take advantage of the pinciple of supeposition. Fo the fist chage, thee is no eneg equied to bing it in fom infinit since thee ae no othe peeisting chages to ceate a potential thee. When the second chage is bought in, the eneg equied can be witten as the value of the second chage multiplied b the electic potential (at the final location of chage ) due to the fist chage: W 4π q q When the thid chage is bought in, the eneg equied is then the value of chage 3 multiplied b the combined potential due to chages and. We could instead calculate this b adding a collection of tems like the one above, one tem fo each possible paiing of chages. As the book deives in moe detail, ou can wite the net esult as W N qi i i ( ) fo a sstem of N chages. If ou instead conside continuous chage distibutions instead of collections of discete chages, the fomula educes to W E all space dτ This is a paticulal useful fom of the eneg, since it emphasizes the ole of the field in stoing eneg. Conductos Conductos can be descibed as mateials with a suppl of fee chages, whee fee means fee to move with essentiall no effot. If a conducto is placed inside an electic field, the mobile chages will move in esponse to this field. Although in solids, the moving chages ae all negative, we could just as easil assume that positive chages do the moving (obviousl in the opposite diection). Positive chages would move in the diection of the etenal applied field. Of couse, once the move, the oveall chage neutalit of the conducto means that positive chages will collect downsteam of the electic field and negative chages on the othe side of the conducto. These chages ceate thei own electic field, and it will oppose the etenall applied field. This will continue until the

22 PHYS 33 electic field within the conducto is zeo. If it wee not zeo, chages would move until the cancelled it out. The lack of an field in the conducto means thee can be no net chage densit within the conducto. Additionall, if the field is zeo evewhee, the potential must be the same evewhee. Note that the potential does not have to be zeo, meel constant. If the conducto does ca an oveall chage (i.e., if chage has been added to a neutal conducto), that chage must sit on the suface fo the easons outlined above. Finall, the electic field has to be pependicula to the suface of the conducto. We can undestand this fom multiple viewpoints; if the conducto is an equipotential, we know that field lines ae pependicula to equipotentials. We could also ecognize that the tangential electic field must be continuous above and below the suface as shown befoe, and thee can be no electic field inside the conducto, so the tangential E field outside the conducto is also zeo. Finall, the pesence of a suface chage gives a discontinuous jump in the nomal component of the electic field; if it s zeo inside the conducto, it has to be nonzeo outside. This means that if we have a chage inside a cavit inside a conducto, thee must be a chage induced on the cavit walls. This is easiest to undestand if we use a spheical conducto with a (concentic) spheical cavit, although it is not at all equied In the dawing above, the dot at the cente is a point chage (call it q). The white egion aound it is the cavit, and the ge oute egion is the conducto. Since thee can be no electic field in the conducto, we choose ou Gaussian suface to be a sphee that is infinitesimall lage than the cavit. Being inside the conducto, we know that E will be zeo, so the total chage inside that egion (cavit a tin faction of the conducto) must be zeo. The onl wa this can happen is if thee is a suface chage on the inteio of the cavit, and if it is equal to q. If the conducto itself is neutal oveall (befoe the intoduction of the point chage in the cente), thee must be a chage of q pesent somewhee in the emaining pat of the conducto. Again, we know it can onl be on the suface, so integating the suface chage densit all ove the conducto s oute suface will give us a value of q. This means that fo a Gaussian suface lage than the conducto, we will get a field that is indistinguishable fom the field of the point chage at the cente b itself. The conducto has done nothing to shield it fom the etenal wold. Capacitance Two neab conductos with chages of opposite sign will of couse ceate an electic field pointing fom the positive conducto to the negative one. The pesence of an electic field

23 PHYS 33 indicates a change in electostatic potential between the two conductos. If the chage on the conductos incease, the potential diffeence between them will also incease (popotionall, in fact). The constant of popotionalit is called the capacitance and is measued in faads. The faad is an etemel lage unit of capacitance; a one-faad capacito could sepaate one coulomb of chage unde a potential diffeence of just one volt. The equation elating these is C Q The capacitance of a paallel-plate capacito is found b assuming the plates ae infinite. The chage on each plate speads out and, as we can find using the pinciple of supeposition, the field between the plates will be Q/(A ) σ/. The potential diffeence between the plates is E dl whee the path connects the two plates. Because of the path independence of this integal and the unifomit of E, this integal becomes just Ed. Using what we just found fo E, we can wite Ed (Q d)/(a ). This gives (again, fo the paallel plate capacito) C (A )/d. The wok done chaging the capacito is q C Q C dw dq dq C Mathematical Techniques Solutions to Laplace s equation ae ve impotant in electostatics. In egions whee chage is pesent, of couse, the impotant equation would be Poisson s equation. We can etact some geneal mathematical esults fom Laplace s equation, howeve. In one dimension, it would simpl be d d The solution to this equation is a potential of the fom mb whee m and b ae constants. In othe wods, the potential lineal inceases as ou move towads positive (o, if m<, negative) values of. Lage values of m coespond to stonge electic fields. Diffeent values of b epesent diffeent choices fo the zeo of potential. Laplace s equation means that thee can t be a minimum o maimum in the potential ecept the ones at the ends of the line. Thee ae no local minima o maima onl the global ones 3

24 PHYS 33 at each end. The linea elationship also means that the potential at an point is the aveage of the potentials at two points equidistant fom the fist point: () ½ ((a) (-a)). If ou emembe, in, this is what the potential between the plates of an ideal paallel-plate capacito looks like. It inceases lineal fom the negative plate to the positive plate, and the electic field is unifom between the plates. This shouldn t be a supise, since the oiginal paallel plate capacito we used in had a vacuum between the plates, so ρ b definition and Poisson s equation educes to Laplace s equation. In two dimensions, Laplace s equation now takes the fom The same basic esults ca ove fom one-dimension. Thee ae again no local maima o minima. The value of at an point is the aveage of its values on a cicle centeed on that point: dl π R (, ) In thee dimensions, at some point is the aveage of the values of on the suface of a sphee centeed on that point: da 4π R ( ) and we still have the esult that the eteme values of occu at the bounda of the egion unde consideation. One of the facts about electostatics that will make ou stud of it easie is the fact that if we can find a function (,,z) that matches the specified bounda conditions (in othe wods, takes the specified values on the suface of some volume), it has to be the coect solution, and thee ae no othe possibilities. You book has a shot and succinct poof of this. It stats b assuming that thee ae two diffeent foms of that satisf Laplace s equation on the bounda. We can wite this as and We can now take the Laplacian of the diffeence between the two potentials 3 3 4

25 PHYS 33 But, since and have to be the same on the bounda (one of ou initial estictions), 3 has to be equal to zeo on that bounda. Of couse, if 3 is zeo evewhee on the bounda (which must contain both the minimum and maimum values of 3 ), we have that 3 evewhee inside the bounda can be both no lage than zeo and no smalle than zeo 3. But, if 3, and we don t eall have two solutions anwa just one. The same agument as above can be made fo Poisson s equation. If thee is a that matches the equied bounda conditions, and a that also matches them, we can again find the Laplacian of the diffeence. This would give: 3 ρ ρ Fom hee, the poof is the same as befoe. 3 satisfies Laplace s equation, and if it is zeo on the bounda of the chage distibution, it must be zeo evewhee inside that bounda as well. We can now sa that, as a geneal ule, the potential inside the bounda is uniquel detemined b the value of potential on the bounda. Image Chages One of the classic poblems in electostatics involves finding the potential due to a point chage above a gounded conducting plane (infinite in size). Since the plane is a gounded conducto, the potential is zeo evewhee on the plane. Above the plane, we might t to descibe the potential b finding kq/ whee is the distance fom the chage to the point of inteest. The poblem with this appoach is that it neglects the pesence of negative chages dawn fom gound into the plane b thei attaction to the positive chage. You can see that this poblem has the potential to get ve complicated ve quickl. The uniqueness theoem can save ou hee, though. If we can find an aangement of chages that makes the potential zeo all along the infinite plane, the will also be guaanteed to give us the coect answe fo the potential in the egion above the plane. One eas wa to do that is to imagine a chage of size q on the opposite side of the plane fom the eal chage q. It should be as fa fom the plane as the eal chage is. In othe wods, think of the gounded plane as being like a lage flat mio. An image of the eal chage will appea in the mio, and it will be as fa fom the mio s suface as the eal chage is. It should be clea fom the smmet of the poblem that eve point on the plane between the two chages is the same distance fom each chage. Assume that the eal chage has,,z coodinates (,,a) while the plane is at z. The image chage (which is not eal, but athe is an atificial wa to simplif the math) theefoe has coodinates (,,-a). The potential at an point (,,z) on the z> side of the plane (whee the eal chage eists) is then found fom 5

26 PHYS 33 (,, z) 4 π q q ( ) ( ) z a z a Fo an point on the plane (whee z) the epession above is cleal zeo. Also, as ou move ve fa awa (even when not on the plane), the potential appoaches zeo. Incidentall, ou should ask ouself If the plane is infinite, what eactl does ve fa awa mean?. The onl emaining thing to set the scale of this poblem is the distance fom the chage to the plane, which we called a. If the distance fom the oigin is much geate than a, we can sa we ae at a point ve fa awa fom the chage. The suface chage densit can be found using one of the ealie esults, namel which in this case is σ n σ (, ) π q a ( a ) 3 / The total chage is found b witing this in clindical coodinates and integating this ove the, plane (which is of couse the location of the gounded conducting plane). The suface element on that plane is da d dθ whee ( ). We can then wite Q π π q a ( a ) 3/ d dθ q While the foce on the eal chage is found easil b the method of images to be just F 4π ( a) q One diffeence between the image chage situation and ealit is in the wok equied to assemble the two sstems. If both chages wee eal, the fist one could be bought in fom infinit fo fee (since thee would be no othe chages pesent to ceate a potential), but this chage would attact the second (opposite) chage bought in fom infinit. The potential ceated b the fist chage at the site of the second would be 6

27 PHYS 33 4 π q a and the wok equied to bing in the second chage fom infinit would just be q o W 4 π q a The poblem is that this is not the coect answe fo ou eal phsical situation consisting of a single chage and a gounded conducting plate. This shouldn t be much of a supise since we ve alead discussed the fact that the electic potential, field and foce all obe the pinciple of linea supeposition, but the potential eneg does not. The coect answe in the chage gounded conducting plane case is half of this. It s a little easie to see wh if we use anothe method of calculating the eneg of the aangement (seen ealie) W E all space dτ In the two chage case, the integal above would be taken ove all space; in the chage plane case, we e esticted to the half of space that contains the actual chage. Anothe common eample using image chages is detailed in ou book; a chage q is nea a gounded conducting sphee of adius R. If the cente is a awa fom the chage (and a > R, so the chage is outside of the sphee), it tuns out that this situation can be duplicated (again in the egion outside the sphee) if the sphee is eplaced b a chage q -(Rq/a) that is located a distance b R /a fom the oigin (fomel occupied b the sphee s cente). The potential in this case at a distance fom the oigin is q q' ( ) 4π ξ ξ whee ξ and ξ ae the distances to the point fom the chages q and q. One of ou homewok poblems will make this moe clea. Laplace s Equation & Sepaation of aiables One wa to solve Laplace s equation fo functions of thee vaiables is to estict ouselves to solutions that ae poducts of functions of one vaiable each. As outlined in ou book, this means that an acceptable solution (in thee dimensions) would take the fom (,, z ) X ( ) Y ( ) Z ( z ) ' 7

28 PHYS 33 The advantage to solutions of this fom is shown below: z Y ( ) Z ( z ) X ( ) X ( ) Z ( z ) Y ( ) X ( ) Y ( ) Z z ( z ) Dividing evewhee b X()Y()Z(z) leaves ou with X ( ) d X d ( ) Y ( ) d Y d ( ) Z ( z ) d Z dz ( z ) (the patial deivatives have now been eplaced b odina deivatives). The ke esult of this method (used ove and ove again in phsics) is that each of these tems is a function of onl one vaiable, and thei sum is constant (zeo in this case). This can onl happen if each tem is independentl equal to some constant value, and the sum of all the constants equals zeo. One of the poblems in ou book looks at the use of this method to solve a poblem whee a metal pipe is infinitel long in the diection but has sides a ( diection) and b (z diection), all fou of which ae gounded. The end of the pipe is held at a specified potential (,z) while the potential goes to zeo as. X d X d C Y d Y d C Z d Z dz C 3 The sum of the C s has to equal zeo, so eithe two will be negative and one positive o two positive and one negative (unless all ae zeo, which is the tivial solution). The bounda conditions detemine what sign each constant must have. Since the potential has to be zeo on the and a sides, the constant C has to be negative so that the solution will be sinusoidal. We get the same esult fo the constant C 3. This means C has to be positive, which woks out well since it has to die off to zeo as, so a sinusoidal function won t wok, and it equies a decaing eponential. You book uses the following foms fo the constants: C The geneal solutions ae then k l C k C3 l 8