MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Depatment Physics 8.07: Electomagnetism II Septembe 5, 202 Pof. Alan Guth PROBLEM SET 2 DUE DATE: Monday, Septembe 24, 202. Eithe hand it in at the lectue, o by 6:00 pm in the 8.07 homewok boxes. READING ASSIGNMENT: Chapte 2 of Giffiths: Electostatics. GRADING OF THIS PROBLEM SET: The poblem set is woth 85 points plus 20points exta cedit. It is theefoe time to claify the opeational definition of exta cedit. We will keep tack of the exta cedit gades sepaately, and at the end of the couse I will fist assign povisional gades based solely on the egula cousewok. I will consult with Pof. Chen and Ahmet Demi, and we will ty to make sue that these gades ae easonable. Then I will add in the exta cedit, allowing the gades to change upwads accodingly. Finally, we will look at each student s gades individually, and we might decide to give a highe gade to some students who ae slightly below a bodeline. Students whose gades have impoved significantly duing the tem, and students whose aveage has been pushed down by single low gade, will be the ones most likely to be boosted. The bottom line is that you should feel fee to skip the exta cedit poblems, and you will still get an excellent gade in the couse if you do well on the egula poblems. Howeve, if you ae the kind of student who eally wants to get the most out of the couse, then I hope that you will find these exta cedit poblems challenging, inteesting, and educational. (As descibed in the solutions to Poblem Set, the poblem sets in the couse will not be gaded in full, but instead only selected poblems will actually be gaded. The exta cedit poblems will neve be among the gaded poblems.) PROBLEM : THE LAPLACIAN AS THE ANTI-LUMPINESS OPERA TOR (5 points) In this poblem you will pove a elation that was stated in lectue. Le ϕ(f) be any scala function of position f. We ae inteested in elating the value of ϕ at an abitay point f 0 to the aveage value of ϕ on a sphee that is centeed at f 0. While the point f 0 is abitay, we can simpify ou notation by choosing a coodinate system so that f 0 is the oigin f0. Then the elation to be poved can be witten ϕ(f0) ϕ (R) = d 3 x 2 ϕ. (.) 4π <R R Hee ϕ(r) epesents the aveage value of ϕ on the suface of a sphee of adius R, which can be witten explicitly as 2π π ϕ (R) = ϕ(r, θ, φ) sin θ dθ dφ. (.2) 4π 0 0
8.07 PROBLEM SET 2, FALL 202 p. 2 The integation in Eq. (.) is ove the volume of a sphee of adius R, centeed at the oigin. The elation to lumpiness can be seen by thinking of ϕ as the density of a pudding. The equation implies that if 2 ϕ = 0, then the value of ϕ at the oigin is the same as the aveage value of its suoundings (no lumpiness). But if 2 ϕ<0, then the value of ϕ at the oigin is highe than the aveage value of its suoundings (i.e., thee is a lump). (a) Defining g() (.3) R fo compactness, use the divegence theoem to show that d 3 x f g ϕ f =0. (.4) <R (b) Use index notation (i.e., f eˆi i ) to show that fo abitay scala functions g(f) and ϕ(f) (g f f ϕ)= g f ϕ f + g 2 ϕ. (.5) (c) Use the identity in pat (b) to ewite the integand of the integal of Eq. (.4), and evaluate each tem sepaately. We suggest that the integal of f g f ϕ be expessed in spheical pola coodinates. Show that the vanishing of the integal in Eq. (.4), e-expessed in this way, implies Eq. (.). PROBLEM 2: CAPACITANCE OF A CYLINDRICAL CAPACITOR (0 points) A vey long conducting cylinde (length e and adius a) caying a total chage +q is suounded by a thin conducting cylindical shell (length e and adius b) with total chage -q, as shown in coss section in the sketch. (a) Using Gauss s law, find an expession fo the electic field E f (f) at points a<<b. Neglect end effects due to the finite length of the capacito. (b) Using you expession fo E f fom pat (a), find the potential diffeence V between the oute shell and the inne cylinde. (c) Deive an expession fo the capacitance of this capacito in tems of the quantities given. What is the capacitance pe unit length? (d) Let the gap d = b a between the cylindes be small compaed to the adii, a and b. Show that in this case you answe fo pat (c) educes to that fo a paallel plate capacito (see Giffiths Eq. (2.54) on p. 05).
8.07 PROBLEM SET 2, FALL 202 p. 3 PROBLEM 3: THE ELECTRIC FIELD, POTENTIAL, AND ENERGY OF A UNIFORM SPHERE OF CHARGE (5 points) (a) A unifomly chaged sphee of chage has adius R and total chage Q. Using Gauss s law, calculate the electic field E f (f) eveywhee. (b) Using the electic field you calculated in pat (a), find the electic potential V (f) eveywhee. (c) Using the expession W = 2 0 E f 2 d 3 x, (3.) all space fo the total wok needed to assemble the chage configuation, calculate W using you expessions above. (d) Using the expession W = ρv d 3 x, (3.2) 2 all space calculate W again using you expessions above. PROBLEM 4: CALCULATING FORCES USING VIRTUAL WORK (0 points) Use vitual wok to calculate the attactive foce between conductos in the paallel plate capacito (aea A, sepaation d). That is, use consevation of enegy to detemine how much wok must be done to move one plate by an infinitesimal amount, and then use the value of the wok to detemine the foce. Do you vitual wok computations in two ways: (a) keeping fixed the chages on the plates, and, (b) keeping a fixed the voltage between the plates. PROBLEM 5: MUTUAL CAPACITANCE (5 points) In lectue we discussed elations of the fom n Q i = C ij V j, i,j =, 2,...,n. (5.) j= govening the potentials and chages of n conductos (with the potential taken to be zeo at spatial infinity). (a) Pove that C ij = C ji.[hint: Conside how much enegy is needed to stat with the system unchaged, then add chage Q i to conducto i, and then add chage Q j to
8.07 PROBLEM SET 2, FALL 202 p. 4 conducto j. Then conside stating again with the system unchaged, and pefoming these opeations in the opposite ode. That is, add chage Q j to conducto j, and then Q i to conducto i. Then think about how to use you answes to pove the desied esult.] (b) Conside a two-conducto configuation. Calculate the conventional capacitance C in tems of C,C 2,C 2, and C 22. (c) Conside two concentic spheical conducting shells of adii a and b with a<b. Call the inne shell conducto, and the oute shell conducto 2. Calculate the matix of capacitances C ij and use you esult fom pat (b) to infe the conventional capacitance C. Compae you answe with Example 2. in Giffiths, p. 05. PROBLEM 6: SPACE CHARGE, VACUUM DIODES, AND THE CHILD LANGMUIR LAW (20 points) Giffiths Poblem 2.48 (p. 07). Challenging! Fo pat (e), you can solve the diffeential equation eithe by guessing a solution and showing that it woks, o by finding a fist integal by the same method that is used in mechanics to go fom Newton s 2nd ode equation of motion to the fist ode equation fo the consevation of mechanical enegy. PROBLEM 7: 2 (/) IN THE LANGUAGE OF DISTRIBUTIONS (20 points exta cedit) This poblem will have a longwinded pedagogical intoduction, since it concens an appoach which was discussed in lectue, but is not discussed in the textbook. In Poblem 5 of Poblem Set, you evaluated 2 (/4π) by eplacing / by / 2 + a 2. Afte calculating g a () 2 (/4π 2 + a 2 ), you showed that its integal ove all space is, and that fo any = 0it appoaches 0as a 0. This execise was intended to convey a useful intuition about δ-functions, and about the elation 2 = δ 3 (f). (7.) 4π Howeve, fom the standpoint of a mathematically igoous teatment, thee is a shotcoming to this and all simila teatments of the δ-function as a limit of a sequence of functions. While the sequence of functions leads to eliable intuition, the pecise mathematical pictue is complicated by the odeing of limits. That is, you showed in you poblem set solutions that g a ()d 3 x = fo any a>0, and hence all space lim g a ()d 3 x =. (7.2) a 0 all space
8.07 PROBLEM SET 2, FALL 202 p. 5 Howeve, if we had taken the limit fist, we would have found 0if =0 lim g a () = (7.3) a 0 if =0, and we showed in lectue that the integal of this function, defined as the aea unde the cuve, is in fact zeo. So we cannot quite say that g a () appoaches a δ-function as a 0. Instead, we have to keep in mind the slightly moe complicated pictue in which g a () acts like a δ-function when a is vey vey small, and behaves exactly as a δ-function if we take the limit a 0 afte any integations have been caied out. Since the integal of the function descibed in Eq. (7.3) vanishes, thee is no nomal function that behaves as a Diac δ-function. Thus the δ-function is technically not a function, but athe what the mathematicians call a genealized function,o a distibution. It is eally the concept of integation that is being genealized, and a distibution is the integand of a genealized integal. Stating with functions of one vaiable, we can conside an abitay function ϕ(x). Its integal, ϕ(x)dx, (7.4) maps the function ϕ(x) into a single eal numbe, the value of its integal. It is a linea map, in the sense that [ϕ (x)+ λϕ 2 (x)] dx = ϕ (x)dx + λ ϕ 2 (x)dx, (7.5) whee λ is a constant. A distibution defines a genealized integal, which is an abitay linea map fom the space of smooth test functions ϕ(x) to eal numbes. These test functions ae equied not only to be smooth, but also to fall off apidly at lage values of x.* The distibution that coesponds to a δ-function is the map which takes the function ϕ(x) to ϕ(x 0 ), its value at some paticula point x 0. While thee is no function that behaves as a Diac δ-function, it is pefectly clea that this map fom functions to eal numbes is well-defined. Thinking of this map as a genealization of integation, we can wite it as ϕ(x) δ(x x 0 )dx ϕ(x 0 ). (7.6) * Vaious choices can be made fo the pecise estictions on the space of test functions. A fequently used choice is the space of Schwatz functions, which ae infinitely diffeentiable, and which have the popety that the function and all its deivatives fall off faste than any powe at lage x. The distibutions associated with this definition of smoothness ae called tempeed distibutions.
8.07 PROBLEM SET 2, FALL 202 p. 6 But emembe that the integal sign hee does not descibe the aea unde a cuve; instead it denotes a linea map fom the function ϕ(x) to a eal numbe, and the symbol δ(x x 0 ) indicates the paticula linea map which maps ϕ(x) to its value at x 0,namely ϕ(x 0 ). Mathematically, Eq. (7.6) defines the δ-function, which is defined solely as a pesciption fo a genealized type of integation. The deivative of a distibution is defined so that genealized integation is consistent with the usual pocedue of integation by pats: d dϕ(x) ϕ(x) δ(x x 0 )dx δ(x x 0 )dx ϕ ' (x 0 ), (7.7) dx dx whee ϕ ' (x) dϕ(x)/dx. Note that we do not include any bounday tems, as ϕ(x) is equied to fall off at lage x fast enough to cause any bounday tems to vanish. We ae now eady to evaluate 2 (/) in the language of distibutions. Note that in the language of functions 2 (/) is ill-defined, because / is not diffeentiable at = 0. But we can pomote / to a distibution by defining it as a mapping fom test functions ϕ(f) to numbes, whee the mapping is given by the (odinay) integal ϕ(f) d 3 x. (7.8) Note that even though / is singula at = 0, this integal is pefectly well defined, since in spheical pola coodinates we have d 3 x = d sin θ dθ dφ. (7.9) By defining the deivative of a distibution by integation by pats, as in Eq. (7.7), we can wite the distibution coesponding to 2 (/), which I will call F [ϕ(f)] fo futue efeence: ( ) ( ) 2 ϕ(f) d 3 x = ϕ(f) i i d 3 x (7.0a) ( ) = i ϕ(f) i d 3 x (70b) ( ) = 2 ϕ(f) d 3 x, (7.0c) so ( ) F [ϕ(f)] 2 ϕ(f) d 3 x. (7.)
8.07 PROBLEM SET 2, FALL 202 p. 7 Note that the squae backets used fo the agument of F is a common notation fo a functional, i.e., a function of a function. Hee F maps the function ϕ(f) tothe numbe given by the (odinay) integal on the ight-hand side of Eq. (7.). AT LAST: THE HOMEWORK PROBLEM: (a) Evaluate F [ϕ(f)] (as defined by Eq. (7.)) fo an abitay smooth test function ϕ(f) which falls off apidly fo lage f. Show that F [ϕ(f)] = 4πϕ(f0). (7.2) Since ϕ(f) δ 3 (f) = ϕ(f0), (7.3) Eq. (7.2) is equivalent to witing ( ) 2 = 4πδ 3 (f) (7.4) in the sense of distibutions, which is the esult we seek. [Hint: Although Eq. (7.) is the defining equation, thee is nothing that pevents you fom integating by pats once to etieve the integal in the fom of Eq. (7.0b). Wite this in spheical pola coodinates, and then ty to evaluate it.] (b) Use the language of distibutions to evaluate 2 ln in two dimensions. (See Poblem 6(d) of Poblem Set.)
MIT OpenCouseWae http://ocw.mit.edu 8.07 Electomagnetism II Fall 202 Fo infomation about citing these mateials o ou Tems of Use, visit: http://ocw.mit.edu/tems.