Lesson 8 Ampère s Law and Differential Operators


 Spencer Lyons
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1 Lesson 8 Ampèe s Law and Diffeential Opeatos Lawence Rees 7 You ma make a single cop of this document fo pesonal use without witten pemission 8 Intoduction Thee ae significant diffeences between the electic and magnetic field lines we have studied so fa Electic field lines spead outwad o convege inwad Magnetic field lines alwas fom closed loops Electic field lines natuall epesent the q/ dependence of the electic field of a point chage Magnetic field contous natuall epesent the i / dependence of the magnetic field of a long wie We found that electic field lines could be mathematicall epesented b flu and Gauss s law of electicit followed as a simple consequence It would stand to eason that we would be able to make simila sots of aguments about the magnetic field contous that would lead to anothe useful mathematical elationship And this is eactl what we ae going to do in this chapte 81 Ampèe's Law As ou ecall, a field contou is a set of sufaces which ae evewhee nomal to the field lines Magnetic field contous oiginate on cuentcaing wies The numbe of contous is chosen to be popotional to the cuent flowing in the wie The contous have a diection which is taken to be the diection of the magnetic field at an point on them When thee ae multiple wies, the field nea each wie, the nea field, is the same as it would be fo a single wie Think About It A cuentcaing wie alone in space has pependicula sufaces coming unifoml outwadl fom it As a second, paallel wie with cuent flowing in the same diection is bought neae to it, what happens to the field contous? Does it make an diffeence what the magnitude of the second cuent is? What happens when the second wie has cuent flowing in the opposite diection to the fist wie? Let s imagine some pependicula sufaces nea a wie Now imagine a closed loop in space nea the wie The loop can be of an shape o sie; the onl equiement is that it be closed A ubbe band is a good eample of a closed loop that can va in shape and sie Such a loop is called an Ampeian loop, as shown in Fig 81 Note that the 1
2 Figue 81 An Ampeian loop and the field contou of a wie caing cuent out of the sceen Ampeian loop has a diection which we have indicated with an aow We will take the diection of the Ampeian loop to be counteclockwise in ou dawings Now we want to define the concept of net numbe of sufaces cossed b the loop To do this, we go aound the loop in the diection of the aow As the loop passes though a pependicula suface, we eithe add +1 o add 1 to the net numbe of sufaces We add +1 if the loop is geneall in the same diection as the suface, and we add 1 if the loop is in the opposite diection geneall same, we mean moe pecisel that thee is an acute angle between the vecto diection of the suface and the vecto diection of the loop as it passes though the suface In Fig 81, the loop cosses a suface in the +1 sense seventeen times and it passes though a suface in the 1 sense once This gives us a net numbe of +16 contous cossed b the Ampeian loop Note that this concept is analogous to the "net numbe of field lines" passing though a Gaussian suface Figue 8 Ampeian loops with cuent passing though them
3 If we take a numbe of diffeent Ampeian loops aound the same wie, as shown in Fig 8, the net numbe of sufaces cossed b each loop is alwas +16 Think About It How would these esults diffe if the cuent in the wie wee going into the sceen? If the cuent wee going into the sceen, the aow on each pependicula suface would be evesed This would in tun change eve +1 into a 1 and eve 1 into a +1, so the net numbe of sufaces pieced b the loop would be 16 As long as we ae caeful to go aound ou Ampeian loop in a geneal counteclockwise sense, we can establish a convenient sign convention: Sign Convention If the numbe of sufaces pieced b an Ampeian loop is positive, the cuent comes out of the sceen If it is negative, cuent goes into the sceen Figue 83 An Ampeian loop with no cuent passing though it If, howeve, we choose ou Ampeian loop in such a wa that no cuent passes though it, as in Fig 83, the net numbe of pependicula sufaces pieced b the loop must be eo If the cuent in the wies doubles, the numbe of pependicula sufaces also doubles It should be evident that we can genealie these esults to give us a law fo magnetic field contous: 3
4 Ampèe s Law Let a set of pependicula sufaces fom a field contou The net numbe of sufaces pieced b an Ampeian loop is popotional to the cuent passing though the loop Ampèe's law is the thid of Mawell's equations As Gauss's law is a geometical wa of stating Coulomb's law, Ampèe's law is a geometical wa of descibing the magnetic field of cuentcaing wies Late we will find mathematical was of epessing this concept and we will use it quantitativel to detemine the magnetic field of cuentcaing objects with high degees of smmet Fist, howeve, we will appl Ampèe s conceptuall to clindical wies Things to emembe Ampeian loops ae closed loops Ampèe s law: The net numbe of sufaces pieced b an Ampeian loop is popotional to the cuent passing though the loop We alwas go aound Ampeian loops in a counteclockwise diection We also define positive cuent to be out and negative cuent to be in Ampèe s law is equivalent to the elation i /(π 8 Cuent Densit Even as it was necessa to define chage densit in the electical case, it is useful to define cuent densit in conjunction with magnetic fields To define cuent densit, let us think of cutting a wie pependiculal to its length and placing a gate in the wie The gate isn t a hole in the wie o a phsical object; it s just a loop though which electons pass We then just count the numbe of electons passing though this gate The cuent passing though the wie is equal to: Cuent (chage pe electon (numbe of electons passing though the gate in one second Now let us make the gate smalle than the coss section of the wie, as in Fig 84 The cuent densit is then: Cuent Densit (chage pe electon (numbe of electons passing though the gate in one second (coss sectional aea of the gate 4
5 small gate Figue 84 A wie with a small gate of aea da We use the lette j to denote cuent densit and measue cuent densit in units of ampees / squae mete ( A / m Think About It Descibe in wods the following cuent densit: 3 sinθ, 1 j(, θ, othewise To find the total cuent passing though a wie, we can integate ove the cosssection of the wie in much the same wa that we integated ove a suface chage densit to find the total chage on a cicula disk That is: I j da Things to emembe: Cuent densit j is the cuent pe unit aea passing though a wie Cuent densit ma va fom egion to egion in a wie; howeve, in tpical wies, cuent densit is quite unifom We can integate cuent densit to get total cuent I j da 83 The Fields of Cuent Distibutions with Radial Clindical Smmet Conside a wie with a cuent densit having adial clindical smmet Recall that b ou definition of adial smmet, the cuent densit ma va with distance fom the ais of the wie,, but it ma not va with angle ecause of this smmet, the sufaces of the field contou outside the wie must come adiall outwad fom the ais and be unifoml spaced Theefoe, the magnetic field in this egion must be the same as the field of a thin wie having 5
6 the same total cuent flowing though it Thus the magnetic field outside a wie does not depend upon the wie's adius ut how do we find the magnetic field within the wie having a adiall smmetic cuent densit? We can be guided b the method we used to find the electic field inside a spheical chage distibution Thee, we divided the spheical chage into two egions: a hollow sphee and a coe We used smmet along with Gauss s law to pove that the electic field inside the hollow sphee is eo Finall, we concluded that the electic field was just the electic field of the coe which in tun was the electic field of a point chage having the same chage as the coe alone Now let s appl the same logic to cuent in a wie Fist, we constuct an Ampeian loop at a adius within the wie We then divide the wie into two pieces, a hollow clinde, and a clindical coe, as shown in Fig 85 Fist, let s conside the hollow coe Since no cuent is passing though the coe, Ampèe s law tells us that the net numbe of pependicula sufaces pieced b the Ampeian loop must be eo ecause of the adial smmet of the hollow wie, an sufaces within the hollow pat of the wie must be adial and the all must point in the same diection (as the sufaces in Fig 81 do Howeve, this contadicts the conclusion that the Ampeian loop pieces eo net sufaces unless thee ae no sufaces at all in the wie We theefoe conclude that thee can be no magnetic field inside a hollow wie that has a clindical cuent densit The entie magnetic field at adius must then be the magnetic field poduced b the cuent in the coe, the cuent that passes though the Ampeian loop of adius Figue 85 Dividing a wie into a hollow conducto and a clindical coe This leads us to the conclusion that ienc ( π whee: ( is the magnetic field at of a wie with a chage densit that is adiall smmetic is the adius of an Ampeian loop o R i enc is the total cuent passing though the Ampeian loop ienc j ( π d 6
7 Things to emembe: The magnetic field is eo inside a hollow wie with adiall smmetic cuent densit The magnetic field ( inside a solid wie with adiall smmetic cuent densit is the magnetic field of the coe (the pat of the wie with adius < alone 84 The Magnetic Line Integal In the same wa that we used flu to ceate an integal fom fo Gauss's law, we now want to do something simila fo the magnetic field to develop an integal equation fo Ampèe's law We peviousl epessed Ampèe's law in the following manne: "The net numbe of pependicula sufaces pieced b an Ampeian loop is popotional to the cuent passing though the loop" We built the geomet of the magnetic field into the field contous in such a wa that the numbe of pependicula sufaces pe unit length is popotional to the magnetic field stength We must now put this in mathematical fom Let us begin with the magnetic field of an infinitellong, cuentcaing wie The magnetic field lines pass aound the wie in concentic cicles and the field contou is composed of a set of halfplanes oiginating on the wie Let s use a magnetic field line at adius as ou Ampeian loop We then know that numbe of sufaces cossed l whee l π is the length of the loop We ma solve this fo the numbe of pependicula sufaces cossed b the loop: Let us epess this as an equalit: numbe of pependicula sufaces cossed l numbe of pependicula sufaces cossed kl whee k is a constant that depends on how man sufaces we associate with each ampee of Λ kλ cuent We call the quantit l the "magnetic line integal" and denote it with an uppe case lambda, Λ We know then that numbe of pependicula sufaces cossed Since l, the line integal is a measue of how much field lies along the Ampeian loop The units of the line integal ae Tm (tesla metes The Magnetic Line Integal fo Segment of a Field Line (81 Λ l 7
8 In a moe geneal case, the Ampeian loop need not lie along a magnetic field line To calculate how much of the field tends to lie along the loop, we need to take the dot poduct of the magnetic field with the diection of the loop: l The poblem we now have is that the elative oientation of and l usuall changes as we go aound the loop We need then to look at the contibution to the total line integal fom one small segment of the wie: dλ d l The total line integal aound the loop is then just the sum ove all such contibutions The Magnetic Line Integal (8 Λ d l The smbol denotes an integation aound the entie Ampeian loop This tpe of integal is called a "line integal" o "path integal We have peviousl used line integals when we have calculated the wok done b a foce The magnetic line integal diffes fom othe onedimensional integals in that it is a sum of the integand along a path which is alwas closed and often iegulal shaped That is, it usuall cannot educe to a simple integal ove d, fo eample In geneal we have to slice the path into small sections, calculate fo each section, calculate the length of the path segment and the diection of the path to get d l, take the dot poduct of these two vectos, and then continue this pocess fo each segment aound the entie path As ou ma be hoping, we will alwas evaluate the line integal in cases whee it educes to a simple fom; howeve, this pesciption fo doing a line integal can be applied to numeical calculations fo integals that ae not easil done b hand The ke to undestanding the behavio of the line integal is to think of the dot poduct between the field and each segment d l of the loop If the path is paallel to the loop d Λ + dl, if the path is opposite the loop dλ dl, and if the path is pependicula to the loop d Λ Things to emembe: The line integal is a quantit that is popotional to the numbe of sufaces (belonging to a field contou pieced b a line segment When we choose a magnetic field line fo the line segment and the magnetic field is constant on the field line, the line integal educes to Λ l The geneal fom fo the line integal is Λ d l 8
9 85 The Integal Fom of Ampèe's Law Now that we have a mathematical epession fo the line integal, we ma easil fomulate Ampèe's law in tems of it Since the line integal is popotional to the net numbe of pependicula sufaces cossed b the Ampeian loop, we know it is also popotional to the cuent flowing though the loop Thus: Λ ci enc whee c is a constant, and i enc is the total cuent passing though the loop To evaluate the constant, let us appl Ampèe's law to the case of a simple cuentcaing wie Since we know the magnetic field of such a wie, we can diectl evaluate the line integal We again use a magnetic field line at adius as the Ampeian loop The line integal is eas to evaluate because the magnetic field is paallel to the path evewhee Theefoe d l is just dl We then can put the epession fo the magnetic field of a long wie into the integal: i Λ dl l π i π Λ ut, b Ampèe's law c i enc Compaing the two esults, we see that the constant c Knowing this constant, we can wite Ampèe's law in its integal fom: Ampèe s Law (83 Λ d l ienc whee: Λ is the magnetic line integal aound an abita Ampeian loop It has units of tesla metes (Tm d l is the (vecto length of a small segment of the Ampeian loop in units of m is the magnetic field on d l in units of tesla (T 7 is the pemeabilit of fee space, 4π 1 Tm / A i is the cuent passing though the Ampeian loop It has units of ampees (A enc As with Gauss's law, Ampèe's law is alwas tue, even if we do not know how to evaluate the integal If a cuent i passes though a loop of an shape o sie, the line integal is alwas i If ou ae asked to find a line integal aound a loop, ou need not evaluate a complicated integal as long as ou know the total cuent passing though the loop 9
10 Think About It When we discussed Ampèe's law ealie, we defined the net numbe of pependicula sufaces intesected as the "numbe of pependicula sufaces pieced in the same geneal diection as the loop minus the numbe cossed in the opposite diection" How does ou definition of the line integal incopoate this concept? In woking with line integals, we appl the same conventions we did ealie: Go aound the Ampeian loop in a counteclockwise diection Cuents coming out of the page have positive line integals; those going into the page have negative line integals To use Ampèe's law to find magnetic fields thee ae two conditions that must be met: We must choose the Ampeian loop such that the magnetic field is paallel to the loop ( is paallel to l d This allows us to simplif the integand: dl l Note that this implies that the Ampeian loop is chosen to be a magnetic field line The poblem must have smmet such that the magnitude of the magnetic field is constant ove the entie loop (o the entie pat of the loop whee the field is noneo Then the integal becomes Ampèe's law then simplifies to dl dl l Ampèe s Law Pactical Fom (84 i enc l whee: is the magnetic field on an Ampeian loop ( must be a constant l is the length of the entie Ampeian loop 7 is the pemeabilit of fee space, 4π 1 Tm / A o R i enc is the total cuent passing though the Ampeian loop ienc j ( d 1
11 Things to emembe: Ampèe s law in integal fom is In pactical applications this educes to Λ l l jda d jda ienc 86 Appling Ampèe's Law Now we wish to appl Ampèe's law to find magnetic fields fo a few special cases A Cuent Distibutions with Radial Clindical Smmet Let us now find the magnetic field of an infinitel long wie with a adiall smmetic distibution of cuent The magnetic field lines ae cleal cicles centeed on the ais of the wie, so we will use a cicula Ampeian loop of adius Smmet also equies the magnitude of the magnetic field to be constant on the loop, so we ma diectl appl Eq (84: o a ienc (85 of a clindical wie ( j( π d π π whee is the adius of the Ampeian loop (the adius at which we wish to find the field a is the adius of the wie The integal is taken ove the aea bounded b the Ampeian loop Thus, the uppe limit in this integal is dependent on whethe we wish to find the magnetic field inside the wie o outside the wie Eample 81 Magnetic field inside a clindical wie A clindical wie of adius R has a cuent I passing though it Find the magnetic field at adius inside the wie In a standad wie, the chage densit is essentiall unifom, so we can easil solve fo it: I I j A πa total Fom hee, we can find the enclosed cuent in one of two diffeent was: I Geometicall, we know that π π a enc jaenc jπ I i I a 11
12 II We can solve fo the enclosed cuent b eplicitl integating ove the cuent densit: I I π I πa πa a enc jda j d π d π i Putting this in Ampèe s law, we have: ( i π π a I enc I π a Eample 8 Magnetic field outside a poton beam A poton beam of adius a has a cuent densit of magnetic outside the beam j α whee α is a constant Find the We poceed in essentiall the same wa as in Eample 81, ecept fo two things: 1 We ae foced to integate to obtain the enclosed cuent, and the integal ove cuent densit must have a as its uppe limit since the thee ae no potons at >a a 4 3 πα a ienc jda jπ d πα d ( i π πα a π a enc α Infinite Plane of Wies Let us now take an aa of wies stacked one on top of anothe and each caing cuent I out of the sceen as shown in the Fig 86 If the plane of wies continues infinitel, we ma use Ampee's law to find the magnetic field 1
13 3 d P Figue 86 The magnetic field of a plana aa of wies The cuent is out of the sceen Fist we need to qualitativel undestand the field At point P in Fig 86, the total magnetic field is the sum of the magnetic fields fom each wie The field fom wie is diected upwad The field fom wie 1 has components both upwad and to the ight The field fom wie 3 has components both upwad and to the left When we add the thee vectos togethe, the net esult is upwad On the lefthand side of the aa of wies the net field must be downwad Let us then daw an Ampeian loop as shown in the figue The magnetic field will va somewhat along the left and ight sides of the loop; howeve, if the wies ae sufficientl closel spaced, this vaiation is small We will take the field to be a constant along these segments The fields on the top and bottom segments of the loop ma not eo; howeve, we do know that the field on the top must be the same as the field on the bottom, and since we ae tavesing the loop in opposite diections on top and bottom, the net contibution to the line integal fom the top plus the bottom must be eo (In an case, we can let the length of the top and bottom be small enough that l on these segments is small Λ The total path integal is then d The enclosed cuent is the cuent in each wie, i, times the numbe of wies N Ampèe's law then gives: Λ d Ni N 1 ni d 13
14 Think About It If two planes of wies ae paallel to each othe, what is the magnetic field in each egion of space? How does the diection of the cuent in each plane affect ou esult? If thee ae two planes of chage with cuent going in opposite diections, the magnetic field has much in common with the electic field of a capacito On the outsides of the planes, the field is eo between the planes the fields add to give ni ased on the magnetic fields of wies and planes of wies, we ma daw some conclusions as we did with the electic field (The dependence on path length fo the plane includes onl the pats of the path fo which the integal is noneo Souce of Field Dimensionalit of Souce Dependence of Path Length Dependence of Field Line Plane C Solenoid A solenoid is a coil of wie wapped aound a clindical coe We will assume that the solenoid has a length much lage than its adius, so that effects caused b the nonunifomit of the field nea the ends ma be ignoed In some was we can think of a solenoid as simila to two planes of wies with cuent going in opposite diections Ignoing end effects, the field outside the solenoid is eo Thus, a solenoid concentates magnetic field lines inside the coil much as paallelplate capacitos concentate electic field lines between the plates Solenoids ae used in man pactical applications whee lage magnetic fields ae needed Often a pemanent magnet is placed inside the solenoid coil so that when cuent is tuned on in the solenoid, the pemanent magnet is pulled into the solenoid o pushed out of it Such a solenoid can do wok, such as stat a ca engine tuning We ma teat solenoids much as the plane of cuent in the above eample We daw an Ampeian loop of length d with one side placed inside the solenoid and one outside On the outside, thee is no field, so the contibution to the line integal is eo On the top and bottom, the magnetic field inside the solenoid is pependicula to the loop, so the contibution is again eo Hence, the line integal is just the contibution fom the left side of the loop: Λ d 14
15 d 1 3 Figue 87 A solenoid Note that, as with the infinite plane of wies, the field stength does not depend on position; we get eactl the same esult no matte whee we place the left side of the Ampeian loop inside the solenoid Thus the field is not onl concentated inside solenoids, but it is unifom within Appling Ampèe s law, we have: d Ni N i d If we call the numbe of tuns pe unit length in the solenoid n, this becomes: Magnetic Field in a Solenoid (86 ni whee: is the magnetic field anwhee within the solenoid The units ae tesla (T 7 is the pemeabilit of fee space, 4π 1 Tm / A n is the numbe of coils pe mete in the solenoid I is the cuent passing though the solenoid 15
16 D Tous Λ A tous is a solenoid which is fomed in a doughnut shape so that the ends ae joined It has simila chaacteistics to a solenoid, but is somewhat diffeent because the wies ae fathe apat fom each othe on the outside the tous than on the inside smmet, we know the magnetic field lines will be cicula loops within the tous We use one field line of adius fo an Ampeian loop as shown in Fig 87 smmet we know the field must be constant on this loop The path integal is then just π If N is the total numbe of tuns in the tous, Ampee's law gives: Λ π N i (87 of a tous π N i Figue 88 A tous with an Ampeian loop (dotted Things to emembe: e able to use Ampèe s law to find the magnetic field inside and outside a wie with adiall smmetic cuent densit e able to use Ampèe s law to find the magnetic field in a solenoid and in a tous You do not need to memoie the fomulas fo these magnetic fields 87 Finding Fields with Diect Integation I m attaching two sections at the end of this chapte as a bief intoduction into othe applications of electomagnetic theo This section deals with diect integation ove chage and cuent distibutions to find electic and magnetic fields The basic idea hee is that, if we know the location and velocit of all the chages in a egion, we can simpl add up all the contibutions to the field fom these chage to obtain the total electic and magnetic fields We alead applied this sot of method to find the electic and magnetic fields of a cuentcaing wie in Lesson 16
17 In this section we will do essentiall the same thing, but in tems of electic and magnetic fields athe than in tems of theads and stubs and thei foces Fist we ll begin with electic potential We know that the electic potential of a point chage is 1 q V ( 4πε R This equation assumes that the souce chage is located at the oigin of a coodinate sstem With an etended souce, we need to make an adjustment in ou notation The following diagam eplains the smbols we will use P R Figue 89 Integating ove a chage distibution We wish to find the electic potential V( at a point P which is located at coodinates (,, We slice the chage distibution into a lot of small egions, such as the cube shown in Fig 89 The coodinates of the cube ae (,, We then define the vectos fom the oigin to the field point P and to the cube to be : ˆ + ˆ + ˆ ˆ + ˆ + ˆ Now, in ou equation fo the electic potential, the that appeas in the denominato is the vecto fom the chage to the field point This is: R 17
18 In tems of this vecto, the contibution to the total electic potential fom the small volume dv, the cube, is 1 dq 1 ρ dv 1 ρ d d d dv 4πε R 4πε R 4πε R The total electic potential is then 1 ρ( d d d (88 V dv 4πε R whee R R ( ˆ + ( ˆ + ( ˆ ( + ( + ( and the integal is ove the entie chage distibution A simila equation holds fo the electic field of the chage distibution, based on Coulomb s law: 1 de 4πε R dq 3 R 1 ρ( R d d d (89 E( de 3 4πε R Eample 83 The chaged od Find the electic field at a distance fom an infinitellong, thin od of unifom chage densit Geomet fo Eample R d
19 We basicall have to use Eq (89 as a ecipe, but let s caefull wite down each step to be sue evething is coect 1 Choose a field point We want the field at a point located a distance fom the od We can put ou aes an place we want, so let s put the point of inteest on the ais Then ˆ Slice the distibution Rods ae eas to slice we just take slices along the od, in the diection 3 Find the coodinates of the slice We need to emembe to put pimes on all the coodinates associated with the souce so the don t become confused with the coodinates of the field point ˆ 4 Find the vecto fom the slice to the field point R ˆ ˆ 5 Find the magnitude of this vecto ( + 1/ R 6 Find the chage on the slice The length of the slice is d (We alwas pime the vaiables that appea in the slices of the souce distibution, too so that dq λ d 7 Now plug evething in to the equations, using cae to get the limits of integation coect I ve done the integation itself using Maple λ E 4πε λ ˆ 4πε λ ˆ 4πε λ E( πε ( ˆ ˆ ( + ˆ ( 3 / 4 + ( + d d 3 / λ ˆ 4πε λ πε d Note that we can also solve this poblem using Gauss s law: q EA ε enc, λ L E, π Lε 3 / E λ πε 19
20 Eample 84 The shot chaged od Find the electic field and the electic potential at a adial distance fom the end of a thin od of unifom chage densit ad length L Geomet fo Eample 84 Again, we use Eqs (88 and (89 Steps 1 though 6 ae identical to Eample 83 R d L 7 Now plug evething in to the equations, using cae to get the limits of integation coect I ve done the integation itself using Maple λ L V 4πε λ L E 4πε λ L ˆ 4πε λ ˆ 4πε λ 4πε ( + ( ˆ ˆ ( + ( / ( + d 1 d L + L d + L 1/ 3 / λ 4πε ˆ ˆ 4 λ πε acsinh λ πε + L [ L ˆ ( + L ˆ ] L + L L d 3 / The second tpe of diect integation poblem we want to conside is that of finding the magnetic field of a segment of thin wie caing a cuent i To do this, we stat with Eq (17 fo a slowl moving paticle to obtain the magnetic field of a moving point chage 1 1 β s E vs E c c
21 If we let the chage move slowl, the electic field is essentiall just the Coulomb field We also use the epession c 1/ ε to simplif the elationship: 1 q R s ε vs 3 4πε R qsvs R 3 4π R What we eall want now, howeve, is an epession that involves cuent athe than the velocit of a single chage Using a little sleight of hand, we can tansfom this equation as: d dq l dq R d R l dqv R dt dt i dl R d s 4π 3 R 4π 3 R 4π 3 R 4π 3 R Integating ove the wie segment, we get a esult suggested b iot and Savat in the 19 th centu: i dl R (81 The iotsavat Law d 3 4π R whee is the magnetic field of a segment of cuent caing wie, measued in tesla (T 7 is the pemeabilit of fee space, 4π 1 Tm / A i is the cuent flowing though the wie dl is a slice of the wie of length dl and pointing in the diection of the cuent R is the vecto distance fom the slice of the wie to the field point Eample 85 the magnetic field of a wie segment A length of wie etends fom to L along the ais it caies cuent i flowing in the diection Find the magnetic field at a point along the ais R d Geomet fo Eample 85 1 L i
22 Let s again keep tack of the solution step b step 1 Choose a field point We e given ˆ Slice the distibution Slice the wie along the ais 3 Find the coodinates of the slice Again, we put pimes on all the coodinates associated with the souce: 4 Find the vecto fom the slice to the field point R ˆ ˆ 5 Find the magnitude of this vecto ( + 1/ R ˆ 6 Find d l The length of the slice is d and the diection of the cuent is in the diection Theefoe dl d ˆ 7 Take the coss poduct dl R d ˆ ˆ ˆ d ˆ ( ( as ˆ ˆ, ˆ ˆ ˆ 8 Now plug evething in to the equations, using cae to get the limits of integation coect I ve done the integation itself using Maple L d 4π i ˆ 4π i ( 4π i d ˆ ( + L L + L + L 3 / Eample 86 A cuent loop A cuent loop of adius a is placed in the  plane with its cente at the oigin of a coodinate sstem Find the magnetic field on the ais The cuent flows counteclockwise when viewed fom the + ais
23 P i θ a dθ Geomet fo Eample 84 Again, we go step b step 1 Choose a field point We e given ˆ Slice the distibution Slice the wie in sections along its length as shown in the figue 3 Find the coodinates of the slice it s most convenient to wok in pola coodinates hee: 4 Find the vecto fom the slice to the field point R ˆ a cosθ ˆ a sinθ ˆ 5 Find the magnitude of this vecto ( a 1/ R + a cosθ ˆ + a sinθ ˆ 6 Find d l The length of the slice is a dθ (using the fomula fo the length of an ac The diection is just a bit tick We know the diection is pependicula to the vecto, so its slope is the negative ecipocal of the slope of We also note that at the segment shown, the component of d l is negative while the component is positive Thus: dl ad θ sin θ ˆ + ad θ cos θ ˆ 7 Take the coss poduct 3
24 dl R adθ ( adθ sinθ ˆ + adθ cosθ ˆ ( ˆ a cosθ ˆ a sinθ ˆ ( a sinθ ˆ + a sin θ ˆ + a cosθ ˆ + a cos θ ˆ a dθ ( sinθ ˆ + cosθ ˆ + ˆ 8 Now plug evething in to the equation, using cae to get the limits of integation coect The integations themselves ae eas: π ia sinθ ˆ + cosθ ˆ + ˆ d d 3 / 4 θ π ia 4π ( + + ia ( a + 3 / ( a + ˆ π ˆ ( a + 3 / Things to emembe: e able to find integal epessions fo the electic field and the electic potential of staight ods and chaged, cicula loops on the ais e able to find integal epessions fo the magnetic fields of staight wie segments and cicula wie loops on the ais You do not need to know how to evaluate the integals that aise 88 The Diffeential Fom of Gauss s Law and Ampèe s Law Thee is one ve fundamental diffeence between the electic fields of static chages and the magnetic fields of cuents: Electic field lines alwas stat o end on electic chages, o else go off to infinit Magnetic field lines alwas fom closed loops This diffeence is evident in Gauss's law and Ampèe's law Flu is a measuement of the net numbe of field lines passing though a suface If a Gaussian suface contains net chage, thee must be net flu though the suface The magnetic flu though a Gaussian suface must be eo because magnetic field lines fom loops, and eve field line which passes into a Gaussian suface must pass back out again The line integal is a measue of how much a field line tends to fom loops Magnetic fields aound wies have line integals popotional to the cuent The line integals of electic fields fom static chages ae alwas eo We can sa that these electic field spead and these magnetic fields loop The diffeence between speading and looping fields can be descibed b two mathematical concepts: divegence and cul 4
25 A Divegence When light as fom the sun pass though a focusing lens, the fom a small image of the sun We sa the light as "convege" If light fom the sun passes though a defocusing lens, the light as spead out and we sa the as "divege" Wheneve electic field lines ae poduced b finite objects (so we can ignoe special cases such as infinite planes, the electic field lines divege if the object has positive chage and convege if it has negative chage This, howeve, is not what we mean mathematicall when we use the tem divegence efoe we define divegence, let us eview a few concepts about fields The electic field is a vecto field That is, at eve point in space we can define an electic field vecto The electic potential is a scala field At eve point in space, the electic potential, a scala quantit, is defined (The two ae closel elated; the electic field gives the magnitude and diection of the change in the electic potential Divegence is simila to electic potential in that it is a scala field A scala quantit, the divegence, is defined at eve point in space The divegence is a measuement of how much a field diveges fom a given point in space That is, it is a measuement of how man field lines begin o end nea that point We can tell if a field has divegence at a point because the flu though a small suface aound the point will not be eo Thus, to measue divegence, we suound a point with a Gaussian suface and measue the flu though it ut to define the divegence at a point, we must let the Gaussian suface get small We then have b Gauss's law: q 1 ( Φ enc ρ E ρ( dv v as v ε ε ε We use this flu then to define the divegence Letting " div E " epesent the divegence of the electic field: (811 The Definition of Divegence div E lim v Φ E v ρ( ε This ma seem just a bit complicated; howeve, the essential idea is that we define divegence at a point to be the flu pe unit volume though a small Gaussian suface located aound that point Think About It A single point chage is in empt space Whee is the divegence of the electic field eo? Whee is it noneo? What if we eplace the point chage with a unifoml chaged sphee? 5
26 Thus, the divegence of an electic field at point in space is popotional to the chage densit at that point The geate the chage densit, the moe field lines begin o end nea that point The souce of an electic field with divegence is electic chage This equation is, in the end, just a diffeent wa of witing Gauss's law We call it the "diffeential fom" of Gauss's law as the divegence can be epessed in tems of deivatives Thus, this is eall a diffeential equation fo the electic field using the methods of patial diffeential equations, we can solve fo the electic field at eve point in space as long as we know the chage densit at eve point in space (Unfotunatel, it still is difficult! A simila equation can be obtained fo magnetic field; howeve, we know that no magnetic chage eists so the ighthand side of the equation is eo In summa: (81 whee Gauss s Law of Electicit Diffeential Fom div E( ρ( ε div E( is the divegence of the electic field at a point ρ ( is the chage densit at a point in units of ε is the pemittivit of fee space It equals 3 C / m 885 Nm 1 1 C / Gauss s Law of Magnetism Diffeential Fom (813 div ( whee div ( is the divegence of the electic field at a point Cul Divegence is a measuement of how much field speads awa fom (o in towad a point souce Divegence is a scala field ecause the field lines of a point paticle tend to emanate unifoml in all diections, thee is no paticula diection associated with the divegence Cul is a measuement of how much a field loops aound a line Cul is a vecto field because at an point we can measue how much the field loops aound lines pointing in the,, o diections Since the cul is a vecto, we can eithe epess cul at a point in space in tems of thee components o in tems of a magnitude and diection The diection of the cul of the magnetic field is the diection cuent flows at that point Fo now, we will assume that the cuent flows in the diection To measue cul, we cleal must el on the line integal To calculate the cul, we 1 choose a point in space, take the line which passes though the point 6
27 in the diection of the cuent, 3 constuct an Ampeian loop aound the line, and 4 calculate the line integal aound the Ampeian loop To get a meaningful quantit fo the cul; howeve, we must let the Ampeian loop get small If we let a be the aea of the loop, we have b Ampee's law: Λ i j ( da j ( a as a enc Hee, j is the cuent densit with the subscipt simpl emphasiing the fact that the cuent is flowing in the diection We then use this elationship to define the component of the cul of : Definition of Cul Λ lim j ( a a (814 [ cul ] ( This then is the diffeential fom of Ampèe's law It simpl states that electical cuents ae a souce of magnetic fields with cul Notice that it is onl in the egion whee thee is cuent that the cul of the field is noneo Howeve, this diffeential equation can be used to solve fo magnetic fields thoughout all space We ma genealie this to cuents which flow in abita diections b defining a vecto cuent densit which points in the diection of the cuent at a given point in space: Ampèe s Law fo Cuents (815 cul ( j( whee: cul ( is the cul of the magnetic field at 7 is the pemeabilit of fee space, 4π 1 Tm / A j( is the cuent densit at in units of A / m It points in the diection of the cuent C The Gadient Opeato Mathematicall speaking, an opeato is something which does something to something else With that vague of a definition, just about anthing could be consideed an opeato And that is tue Opeatos include multiplication, squae oots, deivatives, etc Cul and divegence ae opeatos Howeve, befoe we discuss the cul and divegence opeatos, let us stat fom a moe fundamental opeato, the gadient opeato The gadient is witten as and often ponounced "del" As with othe opeatos, we need to know what the gadient can act on and what is poduced afte it has acted on something The gadient opeato acts on a scala field and poduces a vecto field The gadient tells how much the scala field changes in each of the thee diections,,, and Theefoe, it is much like a deivative in thee dimensions The gadient opeato in Catesian coodinates can be witten as: 7
28 8 + + ˆ ˆ ˆ Even if ou ae not used to the tem "gadient," ou have used gadient opeatos alead We have seen such an epession when we discussed the elationship between electic field and electic potential In shothand, we can wite: Electic Field is the Gadient of Electic Potential (816 V E,, ( In othe wods: V V V V E + + ˆ ˆ ˆ,, ( What this means, then, is that the electic field is a vecto that tells how apidl and what diection the electic potential deceases D Divegence and Cul as Diffeential Opeatos Divegence and cul can be epessed in tems of the gadient opeato Man times students think of the definition of divegence and cul as these diffeential opeatos; howeve, it is best to emembe that the ae defined in tems of the flu and the line integal The diffeential fom of these opeatos is tpified b the following epessions: cul E E E E div E ˆ ˆ ˆ ˆ ˆ ˆ E Eamples 1 A sphee has unifom chage densit ρ Gauss s law, we can find the electic field inside the sphee:
29 q EA ε 4π enc 3 4π ρ E 3ε ρ E 3ε Since the field is adial, we can wite the electic field in vecto fom as E ρ ρ 3ε 3ε ( ˆ + ˆ + ˆ Then we can take the gadient of the electic field: ρ ρ E ( ε ε which is just the diffeential fom of Gauss s law A wie has unifom cuent densit j Ampèe s law, we can find the magnetic field inside the wie: l i enc π π j Since the field lines cicle the wie, it takes a bit of wok to find the field diection We see that the magnetic field id j / times a vecto of length pointing in the tangent diection In Fig 81, we daw this vecto, assuming cuent goes in the diection We can then wite the magnetic field as: j j ( sinθ ˆ + cosθ ˆ ( ˆ + ˆ j 9
30 sinθ ˆ cosθ ˆ θ Figue 81 Finding a vecto of length in the tangential diection Finall, we can take the cul of the magnetic field: j ( ˆ + ˆ j ˆ j This is just the diffeential fom of Ampèe s law Things to emembe: Φ E The definition of divegence: div E lim v v Λ, The definition of cul: [ cul ( ] lim a a ρ Gauss s law of Electicit: E ε Gauss s law of Magnetism: Ampèe s law: j Gadient opeato: ˆ + ˆ + ˆ E E Divegence opeato: div E E + Cul opeato: cul E + 3
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