30 The Electric Field Due to a Continuous Distribution of Charge on a Line

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1 hapte 0 The Electic Field Due to a ontinuous Distibution of hage on a Line 0 The Electic Field Due to a ontinuous Distibution of hage on a Line Evey integal ust include a diffeential (such as d, dt, dq, etc.). An integal is an infinite su of tes. The diffeential is necessay to ake each te infinitesial (vanishingly sall). f ( ) d is okay, g ( y) dy is okay, and h ( t) dt is okay, but neve wite f (), neve wite g ( y) and neve wite h (t). Hee we evisit oulob s Law fo the Electic Field. Recall that oulob s Law fo the Electic Field gives an epession fo the electic field, at an epty point in space, due to a chaged paticle. You have had pactice at finding the electic field at an epty point in space due to a single chaged paticle and due to seveal chaged paticles. In the latte case, you siply calculated the contibution to the electic field at the one epty point in space due to each chaged paticle, and then added the individual contibutions. You wee caeful to keep in ind that each contibution to the electic field at the epty point in space was an electic field vecto, a vecto athe than a scala, hence the individual contibutions had to be added like vectos. A Review oble fo the Electic Field due to a Discete Distibution of hage Let s kick this chapte off by doing a eview poble. The following eaple is one of the sot that you leaned how to do when you fist encounteed oulob s Law fo the Electic Field. You ae given a discete distibution of souce chages and asked to find the electic field (in the case at hand, just the coponent of the electic field) at an epty point in space. The eaple is pesented on the net page. Hee, a wod about one piece of notation used in the solution. The sybol is used to identify a point in space so that the wite can efe to that point, unabiguously, as point. The sybol in this contet does not stand fo a vaiable o a constant. It is just an identification tag. It has no value. It cannot be assigned a value. It does not epesent a distance. It just labels a point. The chage distibution unde consideation hee is called a discete distibution as opposed to a continuous distibution because it consists of seveal individual paticles that ae sepaated fo each othe by soe space. A continuous chage distibution is one in which soe chage is seaed out along soe line o ove soe egion of space. 67

2 hapte 0 The Electic Field Due to a ontinuous Distibution of hage on a Line Eaple 0- (A Review oble) Thee ae two chaged paticles on the -ais of a atesian coodinate syste, q at and q at whee >. Find the coponent of the electic field, due to this pai of paticles, valid fo all points on the -y plane fo which >. y-ais E is the contibution to the electic field at point (at, y) due to chage q. hage q contibutes E to the electic field at. E E + E E E E + Fist, let s get q E : q (, y) y E -ais Again, fo that fist diaga, ( + y ) and cos ( ) + y k q Substituting both of these into E cos yields: k q E + ( ( ) + y ) ( ) y k q ( ) E + [( ) y ] y y-ais E E E (, y) y E cos E q q -ais E E cos Looking at the diaga at the top of this colun, we see that oulob s Law fo the Electic Field yields: k q E k q E cos It is left as an eecise fo the eade to show that: k q( ) E ( ) + y Since [ ] E E + E, we have: k q ( ) k q ( ) E + + [( ) + y ] [( ) y ] 68

3 hapte 0 The Electic Field Due to a ontinuous Distibution of hage on a Line Linea hage Density Okay, enough eview, now lets conside the case in which we have a continuous distibution of chage along soe line segent. In pactice, we could be talking about a chaged piece of sting o thead, a chaged thin od, o even a chaged piece of wie. Fist we need to discuss how one even specifies such a situation. We do so by stating what the linea chage density, the chagepe-length, λ is. Fo now we ll conside the eaning of λ fo a few diffeent situations (befoe we get to the heat of the atte, finding the electic field due to the linea chage distibution). Suppose fo instance we have a one-ete sting etending fo the oigin to.00 along the ais, and that the linea chage density on that sting is given by: µ λ.56. (Just unde the equation, we have depicted the linea chage density gaphically by dawing a line whose dakness epesents the chage density.) Note that if the value of is epessed in etes, λ will have units of µ, units of chage-pe-length, as it ust. Futhe note that fo sall values of, λ is sall, and fo lage values of, λ is lage. That eans that the chage is oe densely packed nea the fa (elative to the oigin) end of the sting. To futhe failiaize ouselves with what λ is, let s calculate the total aount of chage on the sting segent. What we ll do is to get an epession fo the aount of chage on any infinitesial length d of the sting, and add up all such aounts of chage fo all of the infinitesial lengths aking up the sting segent. d (.00, 0) The infinitesial aount of chage dq on the infinitesial length d of the sting is just the chage pe length λ ties the length d of the infinitesial sting segent. dq λ d Note that you can t take the aount of chage on a finite length (such as 5 c) of the sting to be λ ties the length of the segent because λ vaies ove the length of the segent. In the case of an infinitesial segent, evey pat of it is within an infinitesial distance of the position specified by one and the sae value of. The linea chage density doesn t vay on an infinitesial segent because doesn t the segent is siply too shot. 69

4 hapte 0 The Electic Field Due to a ontinuous Distibution of hage on a Line d (.00, 0) dq λ d To get the total chage we just have to add up all the dq s. Each dq is specified by its coesponding value of. To cove all the dq s we have to take into account all the values of fo 0 to.00. Because each dq is the chage on an infinitesial length of the line of chage, the su is going to have an infinite nube of tes. An infinite su of infinitesial pieces is an integal. When we integate dq λ d we get, on the left, the su of all the infinitesial pieces of chage aking up the whole. By definition, the su of all the infinitesial aounts of chage is just the total chage Q (which by the way, is what we ae solving fo); we don t need the tools of integal calculus to deal with the left side of the equation. Integating both sides of the equation yields:. 00 Q λ d µ Using the given epession λ.56 we obtain µ µ Q. 56 d. 56 d 0 0 µ µ (. 00). 56 (0). 8µ A few oe eaples of distibutions of chage follow: Fo instance, conside chage distibuted along the ais, fo 0 to L fo the case in which the chage density is given by λ λ MAX sin( π ad / L) whee λ MAX is a constant having units of chage-pe-length, ad stands fo the units adians, is the position vaiable, and L is the length of the chage distibution. Such a chage distibution has a aiu chage density equal to λ MAX occuing in the iddle of the line segent. 70

5 hapte 0 The Electic Field Due to a ontinuous Distibution of hage on a Line Anothe eaple would be a case in which chage is distibuted on a line segent of length L etending along the y ais fo y a to y a + L with a being a constant and the chage density given by λ 8 µ y In this case the chage on the line is oe densely packed in the egion close to the oigin. (The salle y is, the bigge the value of λ, the chage-pe-length.) The siplest case is the one in which the chage is spead out unifoly ove the line on which thee is chage. In the case of a unifo linea chage distibution, the chage density is the sae eveywhee on the line of chage. In such a case, the linea chage density λ is siply a constant. Futheoe, in such a siple case, and only in such a siple case, the chage density λ is just the total aount of chage Q divided by the length L of the line along which that chage is unifoly distibuted. Fo instance, suppose you ae told that an aount of chage Q.45 is unifoly distibuted along a thin od of length L Then λ is given by: Q λ L. 45 λ λ. 9 The Electic Field Due to a ontinuous Distibution of hage along a Line Okay, now we ae eady to get down to the nitty-gitty. We ae given a continuous distibution of chage along a staight line segent and asked to find the electic field at an epty point in space in the vicinity of the chage distibution. We will conside the case in which both the chage distibution and the epty point in space lie in the -y plane. The values of the coodinates of the epty point in space ae not necessaily specified. We can call the and y. In solving the poble fo a single point in space with unspecified coodinates (, y), ou final answe will have the sybols and y in it, and ou esult will actually give the answe fo an infinite set of points on the -y plane. The plan fo solving such a poble is to find the electic field, due to an infinitesial segent of the chage, at the one epty point in space. We do that fo evey infinitesial segent of the chage, and then add up the esults to get the total electic field. Now once we chop up the chage distibution (in ou ind, fo calculational puposes) into infinitesial (vanishingly sall) pieces, we ae going to wind up with an infinite nube of 7

6 hapte 0 The Electic Field Due to a ontinuous Distibution of hage on a Line pieces and hence an infinite su when we go to add up the contibutions to the electic field at the one single epty point in space due to all the infinitesial segents of the linea chage distibution. That is to say, the esult is going to be an integal. An ipotant consideation that we ust addess is the fact that the electic field, due to each eleent of chage, at the one epty point in space, is a vecto. Hence, what we ae talking about is an infinite su of infinitesial vectos. In geneal, the vectos being added ae all in diffeent diections fo each othe. (an you think of a case so special that the infinite set of infinitesial electic field vectos ae all in the sae diection as each othe? Note that we ae consideing the geneal case, not such a special case.) We know bette than to siply add the agnitudes of the vectos, infinite su o not. Vectos that ae not all in the sae diection as each othe, add like vectos, not like nubes. The thing is, howeve, the coponents of all the infinitesial electic field vectos at the one epty point in space do add like nubes. Likewise fo the y coponents. Thus, if, fo each infinitesial eleent of the chage distibution, we find, not just the electic field at the epty point in space, but the coponent of that electic field, then we can add up all the coponents of the electic field at the epty point in space to get the coponent of the electic field, due to the entie chage distibution, at the one epty point in space. The su is still an infinite su, but this tie it is an infinite su of scalas athe than vectos, and we have the tools fo handling that. Of couse, if we ae asked fo the total electic field, we have to epeat the entie pocedue to get the y coponent of the electic field and then cobine the two coponents of the electic field to get the total. The easy way to do the last step is to use i, j, k notation. That is, once we have E and E y, we can siply wite: E E i + E j y Such a special case occus when one is asked to find the electic field at points on the sae line as the chage distibution, at points outside the chage distibution. Just to educe the copleity of the pobles you will be dealing with, we liit the discussion to the electic field at a point in the -y plane due to a chage distibution in the -y plane. Thus, E z 0 by inspection and we oit it fo the discussion. 7

7 hapte 0 The Electic Field Due to a ontinuous Distibution of hage on a Line Eaple 0- Find the electic field valid fo any point on the positive ais due a 6.0 c long line of chage, lying on the y ais and centeed on the oigin, fo which the chage density is given by λ As usual, we ll stat ou solution with a diaga: y ais y y ais 0.80 λ y Note that we use (and stongly ecoend that you use) pied quantities (, y ) to specify a point on the chage distibution and unpied quantities (, y) to specify the epty point in space at which we wish to know the electic field. Thus, in the diaga, the infinitesial segent of the chage distibution is at (0, y ) and point, the point at which we ae finding the electic field, is at (, 0). Also, ou epession fo the given linea chage density λ y epessed in tes of y athe than y is: λ y The plan hee is to use oulob s Law fo the Electic Field to get the agnitude of the infinitesial electic field vecto at point due to the infinitesial aount of chage dq in the infinitesial segent of length d y. k dq The aount of chage dq in the infinitesial segent d y of the linea chage distibution is given by dq λ dy 7

8 hapte 0 The Electic Field Due to a ontinuous Distibution of hage on a Line y ais dq λ dy y ais 0.80 λ y Fo the diaga, it clea that we can use the ythagoean theoe to epess the distance that point is fo the infinitesial aount of chage dq unde consideation as: + y k dq Substituting this and dq λ dy into ou equation fo ( ) we obtain k λ dy + y Recall that ou plan is to find E, then E y and then put the togethe using E E i + E j. So fo now, let s get an epession fo E. y Based on the vecto coponent diaga at ight we have cos y ais The appeaing in the diaga at ight is the sae that appeas in the diaga above. Based on the plane geoety evident in that diaga (above), we have: cos + y y ais 74

9 hapte 0 The Electic Field Due to a ontinuous Distibution of hage on a Line Substituting both this epession fo cos ( cos k λ dy fo above ( + y diaga yields: + y ) and the epession we deived ) into ou epession cos fo the vecto coponent k λ dy ( + y ) Also, let s go ahead and eplace λ with the given epession λ y : k y dy ( + y ) Now we have an epession fo that includes only one quantity, naely y, that depends on which bit of the chage distibution is unde consideation. Futheoe, although in the diaga y ais dq λ dy y ais 0.80 λ y it appeas that we picked out a paticula infinitesial line segent d y, in fact, the value of y needed to establish its position is not specified. That is, we have an equation fo that is good fo any infinitesial segent d y of the given linea chage distibution. To identify a paticula d y we just have to specify the value of y. Thus to su up all the s we just have to add, to a unning total, the fo each of the possible values of y. Thus we need to integate the epession fo d E fo all the values of y fo 0.80 to k y ( + y ) dy 75

10 hapte 0 The Electic Field Due to a ontinuous Distibution of hage on a Line opying that equation hee: k y ( + y ) dy we note that on the left is the infinite su of all the contibutions to the coponent of the electic field due to all the infinitesial eleents of the line of chage. We don t need any special atheatics techniques to evaluate that. The su of all the pats is the whole. That is, on the left, we have E. The ight side, we can evaluate. Fist, let s facto out the constants: E k y dy ( + y ) The integal is given on you foula sheet. aying out the integation yields: E ( y + + ) y k + ln y + y E. 000 k ( +. 80) (. 80) ( ( +. 80) ) ( (. 80) ) E. 000 k ) 80 ) 80 ) Substituting the value of the oulob constant k fo the foula sheet we obtain E N ) 80 ) 80 ) Finally we have E N ) 80 ) 80 )

11 hapte 0 The Electic Field Due to a ontinuous Distibution of hage on a Line It is inteesting to note that while the position vaiable (which specifies the location of the epty point in space at which the electic field is being calculated) is a constant fo puposes of integation (the location of point does not change as we include the contibution to the electic field at point of each of the infinitesial segents aking up the chage distibution), an actual value was neve specified. Thus ou final esult E N ) 80 ) 80 ) fo E is a function of the position vaiable. Getting the y-coponent of the electic field can be done with a lot less wok than it took to get E if we take advantage of the syety of the chage distibution with espect to the ais. Recall that the chage density λ, fo the case at hand, is given by: λ y Because λ is popotional to y, the value of λ is the sae at the negative of a specified y value as it is at the y value itself. Moe specifically, the aount of chage in each of the two saesize infinitesial eleents of the chage distibution depicted in the following diaga: y ais y y ais is one and the sae value because one eleent is the sae distance below the ais as the othe is above it. This position cicustance also akes the distance that each eleent is fo point the sae as that of the othe, and, it akes the two angles (each of which is labeled in the diaga) have one and the sae value. Thus the two vectos have one and the sae 77

12 hapte 0 The Electic Field Due to a ontinuous Distibution of hage on a Line agnitude. As a esult of the latte two facts (sae angle, sae agnitude of ), the y coponents of the two vectos cancel each othe out. As can be seen in the diaga unde consideation: y ais y y ais one is in the + y diection and the othe in the y diection. The y coponents ae equal and opposite. ) In fact, fo each and evey chage distibution eleent that is above the ais and is thus ceating a downwad contibution to the y coponent of the electic field at point, thee is an eleent that is the sae distance below the ais that is ceating an upwad contibution to the y coponent of the electic field at point, canceling the y coponent of the foe. Thus the net su of all the electic field y coponents (since they cancel pai-wise) is zeo. That is to say that due to the syety of the chage distibution with espect to the ais, E y 0. Thus, E E i. Using the epession fo E that we found above, we have, fo ou final answe: E N ) 80) 80 ) i

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