Lecture 8 - Gauss s Law

Size: px
Start display at page:

Download "Lecture 8 - Gauss s Law"

Transcription

1 Lectue 8 - Gauss s Law A Puzzle... Example Calculate the potential enegy, pe ion, fo an infinite 1D ionic cystal with sepaation a; that is, a ow of equally spaced chages of magnitude e and altenating sign. Hint: The powe-seies expansion of Log[1 + x] may be of use. Solution Suppose the aay is built inwads fom the left (that is, fom negative infinity) as fa as a paticula ion. To add the next positive ion on the ight, the amount of extenal wok equied equals - k e2 a + k e2 2 a - k e2 3 a + = - k e2 a (1) The tems in paenthesis look emakably simila to the Taylo seies of Log[1 + x] when x = 1. (What is the Taylo seies of Log[1 + x]? d 1 Log[1 + x] = dx 1+x = 1 - x + x2 -, and integating both sides yields Log[1 + x] = x - x2 2 + x3 -, since the ight hand side is a polynomial expansion, it is also the Taylo seies of 3 Log[1 + x]. This Taylo seies is conveging fo -1 < x 1 (with convegence on -1 < x < 1 assued by the altenating seies test)). Theefoe the wok equied to bing this chage in equals - k e2 Log[2], which equals the a enegy of the infinite chain pe ion. The addition of futhe paticles on the ight doesn t affect the enegy involved in assembling the pevious ones, so this esult is indeed the enegy pe ion in the entie infinite (in both diections) chain. The esult is negative, which means that it equies enegy to move the ions away fom each othe. This makes sense, because the two neaest neighbos ae of opposite sign. Note that this is an exact esult! It does not assume that a is small. Getting such a nice closed fom is much moe difficult in 2 and 3 dimensions. Theoy Visualizing the Electic Field (Extended) Electic Field Lines Advanced Section: Escaping Field Lines Math Backgound: What is a Suface Integal? We ae aleady familia with suface integals of scala functions. Fo example, the suface aea of a sphee with adius R centeed at the oigin is given by suface da = 0 2 π 0 π R 2 Sin[θ] dθ dϕ = 4 π R 2 (4) Although we use spheical coodinates in this paticula poblem, we could also have used Catesian coodinates

2 2 Lectue nb Although spheical paticula poblem, (with d a = dx dy) o any othe coodinate system. Moe geneally, suppose we ae given a function f[θ, ϕ] defined eveywhee on the suface of the spheical shell of adius R. What is the integal of f[θ, ϕ] acoss the entie spheical shell? This equies a vey mino modification to the fomula above, namely, suface f [θ, ϕ] da = 0 2 π 0 π f [θ, ϕ] R 2 Sin[θ] dθ dϕ (5) We would need to know the specific function f[θ, ϕ] to explicitly evaluate this integal, but we obseve that the suface aea is meely a specific case of Equation (5) with f[θ, ϕ] = 1 eveywhee on the sphee. Lastly, athe than being given a scala function f[θ, ϕ], we could be given a vecto function F[θ, ϕ] defined eveywhee on the sphee. In such a case, we define the suface integal to be suface F[θ, ϕ] da = suface F[θ, ϕ] da da (6) No need to panic! The ight hand side is simply the same suface integal fom Equation (5), except that now the scala function is the dot poduct of two vectos. The fist vecto F[θ, ϕ] is the function that you ae integating ove the sphee. The second vecto da is a unit vecto of an infinitesimal patch in the diection nomal to the suface. On the sphee, the unit nomal vecto at any point is given by d a =, so that we can wite the suface integal as suface F[θ, ϕ] da da = 0 2 π 0 π F[θ, ϕ] R 2 Sin[θ] dθ dϕ (7) Fo closed sufaces, we take the unit nomal to be pointing outwads by convention. Fo open sufaces, you can abitaily choose between the two possible diections that the unit vecto can point (if you switch, it will flip the sign of the suface integal). Usually, we will be woking in setups whee the unit nomal vecto is obvious. Fo example, if we have a sheet in the x-y plane, then the unit nomal vecto will be z (o -z, you can choose eithe as long as you ae consistent thoughout you suface integal) at all points. Similaly, if we have a cylinde whose axis lies on the z-axis, then the unit nomal vecto at its top cap will be z (emembeing that unit nomal vectos point outwads fo closed sufaces) and -z at its bottom cap. Finally, in cylindical coodinates (ρ, θ, z), the unit vecto will be ρ fo all points along the cuved suface of the cylinde. Guass s Law An incedibly useful and beautiful esult, Gauss s Law is definitely woth memoizing! Hee we wite it fo a discete and continuous chage distibution. The integal E da ove the suface, equals 1 times the total chage enclosed by the suface, E da = 1 j q j = 1 ρ dv (8) Fo a combination of both (fo example, a point chage nea an infinite sheet), the Pinciple of Supeposition tells us that we sum ove the discete chages and integate ove the chage distibutions within ou suface. Example Find the electic field due to an infinite line of chage with unifom chage density λ.

3 Lectue nb 3 Solution By symmety, the electic field must point adially outwads fom the line of chage. Cylinde Show E Show a Out[21]= We can use a cylinde whose axis lies on the line of chage and calculate E d a along this cylinde. Since E points adially, E da = 0 along the (flat) top and bottom of the cylinde, and by adial symmety, E d a = E[] da will be a constant value eveywhee along the cone (whee E[] is the magnitude of the electic field at a distance fom the wie). Theefoe, Gauss's Law yields E da = E[] da = E[] (2 π L) = λ L (9) which yields E[] = λ 2 π (10) Altenatively, we could have computed this value by staight-up integation, choosing the adial component of the electic field

4 4 Lectue nb E[] = - k λ dx x = x /2 k λ x x= x /2 x=- = 2 k λ = λ 2 π This yields the same esult as above, but afte significantly moe wok. A Note about Symmety The key to the last poblem was poving that fo a line of chage lying on the x-axis, the electic field E = E[] points adially away fom the wie. Let s pove this explicitly. If the infinitely long wie lies along the x-axis, then the points on the wie ae given by (x, 0, 0) whee x (-, ). Let us conside the electic field at a point (0, 0, z), and let us call the electic field E = E x, E y, E z. Fist, let us flip the setup along the x = 0 as shown below, which means that evey point (and evey vecto) goes fom (x, y, z) (-x, y, z). Theefoe the electic field will go fom E x, E y, E z -E x, E y, E z, shown by the solid aow dashed aow. Howeve, since evey point (x, 0, 0) on the line of chage goes to anothe point (-x, 0, 0) on the line of chage, and the point of inteest (0, 0, z) (0, 0, z) does not change, the physical setup of the poblem emains exactly the same. Theefoe, the electic field befoe and afte the eflection must be the same, and this is only possible if E x = 0. Similaly, conside a eflection about the y = 0 plane, so that evey point goes fom (x, y, z) (x, -y, z). Theefoe the electic field will go fom 0, E y, E z 0, -E y, E z. Each point on the line of chage will not be changed

5 Lectue nb 5 (x, 0, 0) (x, 0, 0), and similaly the point of inteest will not change (0, 0, z) (0, 0, z), so that once again the physical setup is the same and hence the electic field must be the same. This can only happen in E y = 0. We conclude that at the point (0, 0, z), the electic field point staight up (i.e. adially away) fom the line of chage. We can genealize this esult to find the diection of the electic field at an abitay point (x, y, z) using symmety. Fist, we note that because the wie is infinitely long, tanslating along the x-diection cannot change the electic field, since we could equally well have defined ou oigin at some othe point (X, 0, 0) and the physical setup would be identical to that found above, so that the electic field at (X, 0, z) must point in the z diection. Finally, we use the otational symmety of the setup (i.e. otating ou coodinate system about the x-axis) to find that the electic field at any point (x, y, z) points adially outwad fom the wie in the diection 0, y, z. Poblems Field fom a Cylindical Shell Example Conside a chage distibution in the fom of an infinitely long hollow cicula cylinde (like a long chaged pipe) of adius R. If the cylinde has unifom chage pe unit aea σ, what is the electic field inside and outside the cylinde? Solution The electic field inside the cylinde must be pointing adially about the axis of symmety. Theefoe, using a Gaussian suface of a small cylinde with adius < R and length l placed along the axis of the lage cylinde, we find that because thee is no chage inside the cylindical shell. Thus, E = 0 fo all < R. E (2 π l) = q in = 0 (12) The electic field outside the cylinde is found using a the same Gaussian suface but with > R so that o equivalently E = R E (2 π l) = q in = 2 π R l σ (13) σ (2 π R σ). Since the chage pe unit length on the cylindical shell equals 2 π R σ, E = so 2 π that the electic field outside the cylinde is the same as if all of the chage was concentated at on the axis of the cylinde. These esults ae vey inteesting, but let me point out one subtlety in this esult. If you ask fo the electic field

6 6 Lectue nb vey inteesting, point subtlety you E[] as a function of the adial distance fom the cylindical shell, then stating at and coming towads the axis the answe will be E[] = R σ > R 0 < R This implies that at = R, thee is a discontinuity; the value of E[] σ fom the outside of the cylindical shell and E[] 0 fom the inside of the shell. This begs the question, what exactly is the electic field at = R? As we will see fom the electic field of an infinite plane, the answe will be E[R] = σ, and you ae encouaged to do the explicit calculation youself! Fo now, we note that E[R] is the aveage of the values two limits of E[] fom inside and outside the cylindical shell. Exta Poblem: Field fom a Cylindical Shell, Right and Wong Example Find the electic field outside a unifomly chaged hollow cylindical shell with adius R and chage density σ, an infinitesimal distance away fom it. Do this in the following two ways: 1. Slice the shell into paallel infinite ods, and integate the field contibutions fom all the ods. You should obtain the incoect esult of σ. 2. Why isn't the esult coect? Explain how to modify it to obtain the coect esult of σ. Hint: You could vey well have pefomed the above integal in an effot to obtain the electic field an infinitesimal distance inside the cylinde, whee we know the field is zeo. Does the above integation povide a good desciption of what's going on fo points on the shell that ae vey close to the point in question? 3. Confim that the esult equals σ by staight up integation assuming that the point is a finite distance z away fom the cente of the cylinde. What happens when z < R, z = R, and z > R? Solution 1. Let the ods be paameteized by the angle θ as shown in the diagam below. (14) The width of a od is R dθ, so its effective chage pe unit length is λ = σ (R d θ). The od is a distance 2 R Sin θ 2 fom the point P in question, which is infinitesimally close to the top of the cylinde. Only the vetical component of the field fom the od suvives, and this bings in a facto of Sin θ. Using the fact that the field fom a od is 2 λ, we find that the field at the top of the cylinde is (incoectly) 2 π 2 0 π σ R dθ 2 π 2 R Sin θ 2 Sin θ 2 = σ 2 π 0 π dθ = σ (15) Inteestingly, we see that fo a given angula width of a od, all ods yield the same contibution to the vetical electic field at P (since the ones futhe away fom it contibute at a bette angle).

7 Lectue nb 7 2. As stated in the poblem, it is no supise that this answe is incoect, since the same calculation would supposedly yield the field just inside the cylinde too, whee it is zeo instead of σ. Howeve, this calculation does yield the aveage of these two values. The eason why the calculation is invalid is that it doesn t coectly descibe the field due to ods vey close to the given point, that is, fo ods with θ 0. It is incoect fo two easons. Fist, the distance fom a od to the given point is not equal to 2 R Sin θ. Additionally, the field does not point along the line fom the od to the top of the 2 cylinde. It points moe vetically, so the exta facto of Sin θ which we used to pick out the vetical component 2 isn t valid. What is tue is that if we emove a thin stip fom θ = - ϕ 2 to θ = ϕ (fo vey small ϕ 1) at the top of the cylinde, 2 then the above integal is valid fo the emaining pat of the cylinde. In othe wods, the electic field due to the emaining cylinde will be 2 π-ϕ σ σ since we can make 2 π-ϕ abitaily close to 1. By supeposition, the 2 π 2 π total field of the entie cylinde in this field of σ plus the field due to the thin stip at the top. But if the point in question is infinitesimally close to the cylinde, then the thin stip will look like an infinite plane, the field of which we know is σ. The desied total field is then E outside = E cylinde minus stip + E stip = σ + σ = σ (16) E inside = E cylinde minus stip - E stip = σ - σ = 0 (17) whee the minus sign in E inside comes fom the fact that E stip (like an infinite sheet) points in diffeent diections inside and outside the cylinde. Technical note: We have two infinitesimally small quantities in this poblem: the width ϕ of the stip that we ae emoving fom the cylinde and the infinitesimal distance of the point P fom the cylinde. You may (o at least should) be wondeing which of these two infinitesimally small quantities is smalle. This is an impotant point to keep tack of - if you ae blasé about the matte and simply send both quantities to 0 without egad, you could end up with wong esults! In this poblem, we want fo the distance fom P to the cylinde to be smalle, and indeed that is the case. Fo we fist picked a small ϕ (which we can make as small as we like) and then we consideed a point P which was essentially touching the cylinde but on the outside; you can think of this as once you fix ϕ, you bing P in extemely close to the cylinde (and theefoe extemely close to the thin stip). 3. Oient the cylinde s axis to lie on the y-axis and conside the electic field at the point (0, 0, z).

8 8 Lectue nb Define the distance fom the wie at angle θ to the point (0, 0, z) to be, which by the Law of Cosines equals 2 = R 2 + z 2 λ - 2 R z Cos[θ]. Using the chage density λ = σ (R dθ), the electic field fom an infinite wie, 2 π and the z-component z-r Cos[θ] of the electic field, E = 0 2 π R σ dθ 2 π z-r Cos[θ] = R σ 2 π 0 2 π z-r Cos[θ] R 2 +z 2-2 R z Cos[θ] dθ (18) This integal is not supe-difficult to evaluate, but cae must be taken as to whethe z < R, z = R, o z > R. The full esult is given by E = R σ (1-Sign[R-z]) z (19) Integate R σ z - R Cos[θ], {θ, 0, 2 π}, Assumptions 0 < ϵ0 && 0 < σ && 0 < R && 0 < z 2 π ϵ0 R 2 + z 2-2 R z Cos[θ] R σ (-1 + Sign[R - z]) - 2 z ϵ0 In othe wods, E = 0 z < R σ z = R R z σ z > R We see that when we appoach the suface of the cylinde fom the inside E = 0, wheeas when we appoach it fom the outside, E = R σ σ. The electic field on the actual suface equals the aveage of these two values, z E = σ. (20) Mathematica Initialization

Flux. Area Vector. Flux of Electric Field. Gauss s Law

Flux. Area Vector. Flux of Electric Field. Gauss s Law Gauss s Law Flux Flux in Physics is used to two distinct ways. The fist meaning is the ate of flow, such as the amount of wate flowing in a ive, i.e. volume pe unit aea pe unit time. O, fo light, it is

More information

Gauss Law. Physics 231 Lecture 2-1

Gauss Law. Physics 231 Lecture 2-1 Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing

More information

Physics 2212 GH Quiz #2 Solutions Spring 2016

Physics 2212 GH Quiz #2 Solutions Spring 2016 Physics 2212 GH Quiz #2 Solutions Sping 216 I. 17 points) Thee point chages, each caying a chage Q = +6. nc, ae placed on an equilateal tiangle of side length = 3. mm. An additional point chage, caying

More information

Hopefully Helpful Hints for Gauss s Law

Hopefully Helpful Hints for Gauss s Law Hopefully Helpful Hints fo Gauss s Law As befoe, thee ae things you need to know about Gauss s Law. In no paticula ode, they ae: a.) In the context of Gauss s Law, at a diffeential level, the electic flux

More information

Welcome to Physics 272

Welcome to Physics 272 Welcome to Physics 7 Bob Mose mose@phys.hawaii.edu http://www.phys.hawaii.edu/~mose/physics7.html To do: Sign into Masteing Physics phys-7 webpage Registe i-clickes (you i-clicke ID to you name on class-list)

More information

CHAPTER 25 ELECTRIC POTENTIAL

CHAPTER 25 ELECTRIC POTENTIAL CHPTE 5 ELECTIC POTENTIL Potential Diffeence and Electic Potential Conside a chaged paticle of chage in a egion of an electic field E. This filed exets an electic foce on the paticle given by F=E. When

More information

( ) Make-up Tests. From Last Time. Electric Field Flux. o The Electric Field Flux through a bit of area is

( ) Make-up Tests. From Last Time. Electric Field Flux. o The Electric Field Flux through a bit of area is Mon., 3/23 Wed., 3/25 Thus., 3/26 Fi., 3/27 Mon., 3/30 Tues., 3/31 21.4-6 Using Gauss s & nto to Ampee s 21.7-9 Maxwell s, Gauss s, and Ampee s Quiz Ch 21, Lab 9 Ampee s Law (wite up) 22.1-2,10 nto to

More information

Question 1: The dipole

Question 1: The dipole Septembe, 08 Conell Univesity, Depatment of Physics PHYS 337, Advance E&M, HW #, due: 9/5/08, :5 AM Question : The dipole Conside a system as discussed in class and shown in Fig.. in Heald & Maion.. Wite

More information

B. Spherical Wave Propagation

B. Spherical Wave Propagation 11/8/007 Spheical Wave Popagation notes 1/1 B. Spheical Wave Popagation Evey antenna launches a spheical wave, thus its powe density educes as a function of 1, whee is the distance fom the antenna. We

More information

Continuous Charge Distributions: Electric Field and Electric Flux

Continuous Charge Distributions: Electric Field and Electric Flux 8/30/16 Quiz 2 8/25/16 A positive test chage qo is eleased fom est at a distance away fom a chage of Q and a distance 2 away fom a chage of 2Q. How will the test chage move immediately afte being eleased?

More information

$ i. !((( dv vol. Physics 8.02 Quiz One Equations Fall q 1 q 2 r 2 C = 2 C! V 2 = Q 2 2C F = 4!" or. r ˆ = points from source q to observer

$ i. !((( dv vol. Physics 8.02 Quiz One Equations Fall q 1 q 2 r 2 C = 2 C! V 2 = Q 2 2C F = 4! or. r ˆ = points from source q to observer Physics 8.0 Quiz One Equations Fall 006 F = 1 4" o q 1 q = q q ˆ 3 4" o = E 4" o ˆ = points fom souce q to obseve 1 dq E = # ˆ 4" 0 V "## E "d A = Q inside closed suface o d A points fom inside to V =

More information

working pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50

working pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50 woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50 CHAPTER7 Solid angle, 3D integals, Gauss s Theoem, and a Delta Function We define the solid angle,

More information

Page 1 of 6 Physics II Exam 1 155 points Name Discussion day/time Pat I. Questions 110. 8 points each. Multiple choice: Fo full cedit, cicle only the coect answe. Fo half cedit, cicle the coect answe and

More information

2 E. on each of these two surfaces. r r r r. Q E E ε. 2 2 Qencl encl right left 0

2 E. on each of these two surfaces. r r r r. Q E E ε. 2 2 Qencl encl right left 0 Ch : 4, 9,, 9,,, 4, 9,, 4, 8 4 (a) Fom the diagam in the textbook, we see that the flux outwad though the hemispheical suface is the same as the flux inwad though the cicula suface base of the hemisphee

More information

Today s Plan. Electric Dipoles. More on Gauss Law. Comment on PDF copies of Lectures. Final iclicker roll-call

Today s Plan. Electric Dipoles. More on Gauss Law. Comment on PDF copies of Lectures. Final iclicker roll-call Today s Plan lectic Dipoles Moe on Gauss Law Comment on PDF copies of Lectues Final iclicke oll-call lectic Dipoles A positive (q) and negative chage (-q) sepaated by a small distance d. lectic dipole

More information

Electrostatics (Electric Charges and Field) #2 2010

Electrostatics (Electric Charges and Field) #2 2010 Electic Field: The concept of electic field explains the action at a distance foce between two chaged paticles. Evey chage poduces a field aound it so that any othe chaged paticle expeiences a foce when

More information

Your Comments. Do we still get the 80% back on homework? It doesn't seem to be showing that. Also, this is really starting to make sense to me!

Your Comments. Do we still get the 80% back on homework? It doesn't seem to be showing that. Also, this is really starting to make sense to me! You Comments Do we still get the 8% back on homewok? It doesn't seem to be showing that. Also, this is eally stating to make sense to me! I am a little confused about the diffeences in solid conductos,

More information

EM-2. 1 Coulomb s law, electric field, potential field, superposition q. Electric field of a point charge (1)

EM-2. 1 Coulomb s law, electric field, potential field, superposition q. Electric field of a point charge (1) EM- Coulomb s law, electic field, potential field, supeposition q ' Electic field of a point chage ( ') E( ) kq, whee k / 4 () ' Foce of q on a test chage e at position is ee( ) Electic potential O kq

More information

! E da = 4πkQ enc, has E under the integral sign, so it is not ordinarily an

! E da = 4πkQ enc, has E under the integral sign, so it is not ordinarily an Physics 142 Electostatics 2 Page 1 Electostatics 2 Electicity is just oganized lightning. Geoge Calin A tick that sometimes woks: calculating E fom Gauss s law Gauss s law,! E da = 4πkQ enc, has E unde

More information

Module 05: Gauss s s Law a

Module 05: Gauss s s Law a Module 05: Gauss s s Law a 1 Gauss s Law The fist Maxwell Equation! And a vey useful computational technique to find the electic field E when the souce has enough symmety. 2 Gauss s Law The Idea The total

More information

Physics 181. Assignment 4

Physics 181. Assignment 4 Physics 181 Assignment 4 Solutions 1. A sphee has within it a gavitational field given by g = g, whee g is constant and is the position vecto of the field point elative to the cente of the sphee. This

More information

13. The electric field can be calculated by Eq. 21-4a, and that can be solved for the magnitude of the charge N C m 8.

13. The electric field can be calculated by Eq. 21-4a, and that can be solved for the magnitude of the charge N C m 8. CHAPTR : Gauss s Law Solutions to Assigned Poblems Use -b fo the electic flux of a unifom field Note that the suface aea vecto points adially outwad, and the electic field vecto points adially inwad Thus

More information

An o5en- confusing point:

An o5en- confusing point: An o5en- confusing point: Recall this example fom last lectue: E due to a unifom spheical suface chage, density = σ. Let s calculate the pessue on the suface. Due to the epulsive foces, thee is an outwad

More information

Physics 2B Chapter 22 Notes - Magnetic Field Spring 2018

Physics 2B Chapter 22 Notes - Magnetic Field Spring 2018 Physics B Chapte Notes - Magnetic Field Sping 018 Magnetic Field fom a Long Staight Cuent-Caying Wie In Chapte 11 we looked at Isaac Newton s Law of Gavitation, which established that a gavitational field

More information

Gauss s Law Simulation Activities

Gauss s Law Simulation Activities Gauss s Law Simulation Activities Name: Backgound: The electic field aound a point chage is found by: = kq/ 2 If thee ae multiple chages, the net field at any point is the vecto sum of the fields. Fo a

More information

2. Electrostatics. Dr. Rakhesh Singh Kshetrimayum 8/11/ Electromagnetic Field Theory by R. S. Kshetrimayum

2. Electrostatics. Dr. Rakhesh Singh Kshetrimayum 8/11/ Electromagnetic Field Theory by R. S. Kshetrimayum 2. Electostatics D. Rakhesh Singh Kshetimayum 1 2.1 Intoduction In this chapte, we will study how to find the electostatic fields fo vaious cases? fo symmetic known chage distibution fo un-symmetic known

More information

PHYS 1444 Lecture #5

PHYS 1444 Lecture #5 Shot eview Chapte 24 PHYS 1444 Lectue #5 Tuesday June 19, 212 D. Andew Bandt Capacitos and Capacitance 1 Coulom s Law The Fomula QQ Q Q F 1 2 1 2 Fomula 2 2 F k A vecto quantity. Newtons Diection of electic

More information

Physics 235 Chapter 5. Chapter 5 Gravitation

Physics 235 Chapter 5. Chapter 5 Gravitation Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus

More information

Chapter 23: GAUSS LAW 343

Chapter 23: GAUSS LAW 343 Chapte 23: GAUSS LAW 1 A total chage of 63 10 8 C is distibuted unifomly thoughout a 27-cm adius sphee The volume chage density is: A 37 10 7 C/m 3 B 69 10 6 C/m 3 C 69 10 6 C/m 2 D 25 10 4 C/m 3 76 10

More information

Your Comments. Conductors and Insulators with Gauss's law please...so basically everything!

Your Comments. Conductors and Insulators with Gauss's law please...so basically everything! You Comments I feel like I watch a pe-lectue, and agee with eveything said, but feel like it doesn't click until lectue. Conductos and Insulatos with Gauss's law please...so basically eveything! I don't

More information

Review: Electrostatics and Magnetostatics

Review: Electrostatics and Magnetostatics Review: Electostatics and Magnetostatics In the static egime, electomagnetic quantities do not vay as a function of time. We have two main cases: ELECTROSTATICS The electic chages do not change postion

More information

Math 2263 Solutions for Spring 2003 Final Exam

Math 2263 Solutions for Spring 2003 Final Exam Math 6 Solutions fo Sping Final Exam ) A staightfowad appoach to finding the tangent plane to a suface at a point ( x, y, z ) would be to expess the cuve as an explicit function z = f ( x, y ), calculate

More information

University Physics (PHY 2326)

University Physics (PHY 2326) Chapte Univesity Physics (PHY 6) Lectue lectostatics lectic field (cont.) Conductos in electostatic euilibium The oscilloscope lectic flux and Gauss s law /6/5 Discuss a techniue intoduced by Kal F. Gauss

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Electromagnetism II September 15, 2012 Prof. Alan Guth PROBLEM SET 2

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Electromagnetism II September 15, 2012 Prof. Alan Guth PROBLEM SET 2 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,

More information

Objects usually are charged up through the transfer of electrons from one object to the other.

Objects usually are charged up through the transfer of electrons from one object to the other. 1 Pat 1: Electic Foce 1.1: Review of Vectos Review you vectos! You should know how to convet fom pola fom to component fom and vice vesa add and subtact vectos multiply vectos by scalas Find the esultant

More information

PHY2061 Enriched Physics 2 Lecture Notes. Gauss Law

PHY2061 Enriched Physics 2 Lecture Notes. Gauss Law PHY61 Eniched Physics Lectue Notes Law Disclaime: These lectue notes ae not meant to eplace the couse textbook. The content may be incomplete. ome topics may be unclea. These notes ae only meant to be

More information

1 Spherical multipole moments

1 Spherical multipole moments Jackson notes 9 Spheical multipole moments Suppose we have a chage distibution ρ (x) wheeallofthechageiscontained within a spheical egion of adius R, as shown in the diagam. Then thee is no chage in the

More information

Φ E = E A E A = p212c22: 1

Φ E = E A E A = p212c22: 1 Chapte : Gauss s Law Gauss s Law is an altenative fomulation of the elation between an electic field and the souces of that field in tems of electic flux. lectic Flux Φ though an aea A ~ Numbe of Field

More information

Physics 2A Chapter 10 - Moment of Inertia Fall 2018

Physics 2A Chapter 10 - Moment of Inertia Fall 2018 Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion.

More information

Chapter 7-8 Rotational Motion

Chapter 7-8 Rotational Motion Chapte 7-8 Rotational Motion What is a Rigid Body? Rotational Kinematics Angula Velocity ω and Acceleation α Unifom Rotational Motion: Kinematics Unifom Cicula Motion: Kinematics and Dynamics The Toque,

More information

Chapter 22 The Electric Field II: Continuous Charge Distributions

Chapter 22 The Electric Field II: Continuous Charge Distributions Chapte The lectic Field II: Continuous Chage Distibutions A ing of adius a has a chage distibution on it that vaies as l(q) l sin q, as shown in Figue -9. (a) What is the diection of the electic field

More information

EM Boundary Value Problems

EM Boundary Value Problems EM Bounday Value Poblems 10/ 9 11/ By Ilekta chistidi & Lee, Seung-Hyun A. Geneal Desciption : Maxwell Equations & Loentz Foce We want to find the equations of motion of chaged paticles. The way to do

More information

Chapter 21: Gauss s Law

Chapter 21: Gauss s Law Chapte : Gauss s Law Gauss s law : intoduction The total electic flux though a closed suface is equal to the total (net) electic chage inside the suface divided by ε Gauss s law is equivalent to Coulomb

More information

PY208 Matter & Interactions Final Exam S2005

PY208 Matter & Interactions Final Exam S2005 PY Matte & Inteactions Final Exam S2005 Name (pint) Please cicle you lectue section below: 003 (Ramakishnan 11:20 AM) 004 (Clake 1:30 PM) 005 (Chabay 2:35 PM) When you tun in the test, including the fomula

More information

Chapter 13 Gravitation

Chapter 13 Gravitation Chapte 13 Gavitation In this chapte we will exploe the following topics: -Newton s law of gavitation, which descibes the attactive foce between two point masses and its application to extended objects

More information

Qualifying Examination Electricity and Magnetism Solutions January 12, 2006

Qualifying Examination Electricity and Magnetism Solutions January 12, 2006 1 Qualifying Examination Electicity and Magnetism Solutions Januay 12, 2006 PROBLEM EA. a. Fist, we conside a unit length of cylinde to find the elationship between the total chage pe unit length λ and

More information

arxiv: v1 [physics.pop-ph] 3 Jun 2013

arxiv: v1 [physics.pop-ph] 3 Jun 2013 A note on the electostatic enegy of two point chages axiv:1306.0401v1 [physics.pop-ph] 3 Jun 013 A C Tot Instituto de Física Univesidade Fedeal do io de Janeio Caixa Postal 68.58; CEP 1941-97 io de Janeio,

More information

Objectives: After finishing this unit you should be able to:

Objectives: After finishing this unit you should be able to: lectic Field 7 Objectives: Afte finishing this unit you should be able to: Define the electic field and explain what detemines its magnitude and diection. Wite and apply fomulas fo the electic field intensity

More information

ω = θ θ o = θ θ = s r v = rω

ω = θ θ o = θ θ = s r v = rω Unifom Cicula Motion Unifom cicula motion is the motion of an object taveling at a constant(unifom) speed in a cicula path. Fist we must define the angula displacement and angula velocity The angula displacement

More information

Nuclear and Particle Physics - Lecture 20 The shell model

Nuclear and Particle Physics - Lecture 20 The shell model 1 Intoduction Nuclea and Paticle Physics - Lectue 0 The shell model It is appaent that the semi-empiical mass fomula does a good job of descibing tends but not the non-smooth behaviou of the binding enegy.

More information

15 Solving the Laplace equation by Fourier method

15 Solving the Laplace equation by Fourier method 5 Solving the Laplace equation by Fouie method I aleady intoduced two o thee dimensional heat equation, when I deived it, ecall that it taes the fom u t = α 2 u + F, (5.) whee u: [0, ) D R, D R is the

More information

(Sample 3) Exam 1 - Physics Patel SPRING 1998 FORM CODE - A (solution key at end of exam)

(Sample 3) Exam 1 - Physics Patel SPRING 1998 FORM CODE - A (solution key at end of exam) (Sample 3) Exam 1 - Physics 202 - Patel SPRING 1998 FORM CODE - A (solution key at end of exam) Be sue to fill in you student numbe and FORM lette (A, B, C) on you answe sheet. If you foget to include

More information

Today in Physics 122: getting V from E

Today in Physics 122: getting V from E Today in Physics 1: getting V fom E When it s best to get V fom E, athe than vice vesa V within continuous chage distibutions Potential enegy of continuous chage distibutions Capacitance Potential enegy

More information

AST 121S: The origin and evolution of the Universe. Introduction to Mathematical Handout 1

AST 121S: The origin and evolution of the Universe. Introduction to Mathematical Handout 1 Please ead this fist... AST S: The oigin and evolution of the Univese Intoduction to Mathematical Handout This is an unusually long hand-out and one which uses in places mathematics that you may not be

More information

B da = 0. Q E da = ε. E da = E dv

B da = 0. Q E da = ε. E da = E dv lectomagnetic Theo Pof Ruiz, UNC Asheville, doctophs on YouTube Chapte Notes The Maxwell quations in Diffeential Fom 1 The Maxwell quations in Diffeential Fom We will now tansfom the integal fom of the

More information

Physics 122, Fall September 2012

Physics 122, Fall September 2012 Physics 1, Fall 1 7 Septembe 1 Today in Physics 1: getting V fom E When it s best to get V fom E, athe than vice vesa V within continuous chage distibutions Potential enegy of continuous chage distibutions

More information

16.1 Permanent magnets

16.1 Permanent magnets Unit 16 Magnetism 161 Pemanent magnets 16 The magnetic foce on moving chage 163 The motion of chaged paticles in a magnetic field 164 The magnetic foce exeted on a cuent-caying wie 165 Cuent loops and

More information

PHYS 110B - HW #7 Spring 2004, Solutions by David Pace Any referenced equations are from Griffiths Problem statements are paraphrased

PHYS 110B - HW #7 Spring 2004, Solutions by David Pace Any referenced equations are from Griffiths Problem statements are paraphrased PHYS 0B - HW #7 Sping 2004, Solutions by David Pace Any efeenced euations ae fom Giffiths Poblem statements ae paaphased. Poblem 0.3 fom Giffiths A point chage,, moves in a loop of adius a. At time t 0

More information

Chapter 22: Electric Fields. 22-1: What is physics? General physics II (22102) Dr. Iyad SAADEDDIN. 22-2: The Electric Field (E)

Chapter 22: Electric Fields. 22-1: What is physics? General physics II (22102) Dr. Iyad SAADEDDIN. 22-2: The Electric Field (E) Geneal physics II (10) D. Iyad D. Iyad Chapte : lectic Fields In this chapte we will cove The lectic Field lectic Field Lines -: The lectic Field () lectic field exists in a egion of space suounding a

More information

Exam 1. Exam 1 is on Tuesday, February 14, from 5:00-6:00 PM.

Exam 1. Exam 1 is on Tuesday, February 14, from 5:00-6:00 PM. Exam 1 Exam 1 is on Tuesday, Febuay 14, fom 5:00-6:00 PM. Testing Cente povides accommodations fo students with special needs I must set up appointments one week befoe exam Deadline fo submitting accommodation

More information

Algebra-based Physics II

Algebra-based Physics II lgebabased Physics II Chapte 19 Electic potential enegy & The Electic potential Why enegy is stoed in an electic field? How to descibe an field fom enegetic point of view? Class Website: Natual way of

More information

PHYS 1444 Section 501 Lecture #7

PHYS 1444 Section 501 Lecture #7 PHYS 1444 Section 51 Lectue #7 Wednesday, Feb. 8, 26 Equi-potential Lines and Sufaces Electic Potential Due to Electic Dipole E detemined fom V Electostatic Potential Enegy of a System of Chages Capacitos

More information

Review for Midterm-1

Review for Midterm-1 Review fo Midtem-1 Midtem-1! Wednesday Sept. 24th at 6pm Section 1 (the 4:10pm class) exam in BCC N130 (Business College) Section 2 (the 6:00pm class) exam in NR 158 (Natual Resouces) Allowed one sheet

More information

The Divergence Theorem

The Divergence Theorem 13.8 The ivegence Theoem Back in 13.5 we ewote Geen s Theoem in vecto fom as C F n ds= div F x, y da ( ) whee C is the positively-oiented bounday cuve of the plane egion (in the xy-plane). Notice this

More information

Target Boards, JEE Main & Advanced (IIT), NEET Physics Gauss Law. H. O. D. Physics, Concept Bokaro Centre P. K. Bharti

Target Boards, JEE Main & Advanced (IIT), NEET Physics Gauss Law. H. O. D. Physics, Concept Bokaro Centre P. K. Bharti Page 1 CONCPT: JB-, Nea Jitenda Cinema, City Cente, Bokao www.vidyadishti.og Gauss Law Autho: Panjal K. Bhati (IIT Khaagpu) Mb: 74884484 Taget Boads, J Main & Advanced (IIT), NT 15 Physics Gauss Law Autho:

More information

ELECTROSTATICS::BHSEC MCQ 1. A. B. C. D.

ELECTROSTATICS::BHSEC MCQ 1. A. B. C. D. ELETROSTATIS::BHSE 9-4 MQ. A moving electic chage poduces A. electic field only. B. magnetic field only.. both electic field and magnetic field. D. neithe of these two fields.. both electic field and magnetic

More information

Vectors, Vector Calculus, and Coordinate Systems

Vectors, Vector Calculus, and Coordinate Systems ! Revised Apil 11, 2017 1:48 PM! 1 Vectos, Vecto Calculus, and Coodinate Systems David Randall Physical laws and coodinate systems Fo the pesent discussion, we define a coodinate system as a tool fo descibing

More information

Phys-272 Lecture 17. Motional Electromotive Force (emf) Induced Electric Fields Displacement Currents Maxwell s Equations

Phys-272 Lecture 17. Motional Electromotive Force (emf) Induced Electric Fields Displacement Currents Maxwell s Equations Phys-7 Lectue 17 Motional Electomotive Foce (emf) Induced Electic Fields Displacement Cuents Maxwell s Equations Fom Faaday's Law to Displacement Cuent AC geneato Magnetic Levitation Tain Review of Souces

More information

transformation Earth V-curve (meridian) λ Conical projection. u,v curves on the datum surface projected as U,V curves on the projection surface

transformation Earth V-curve (meridian) λ Conical projection. u,v curves on the datum surface projected as U,V curves on the projection surface . CONICAL PROJECTIONS In elementay texts on map pojections, the pojection sufaces ae often descibed as developable sufaces, such as the cylinde (cylindical pojections) and the cone (conical pojections),

More information

ECE 3318 Applied Electricity and Magnetism. Spring Prof. David R. Jackson ECE Dept. Notes 13

ECE 3318 Applied Electricity and Magnetism. Spring Prof. David R. Jackson ECE Dept. Notes 13 ECE 338 Applied Electicity and Magnetism ping 07 Pof. David R. Jackson ECE Dept. Notes 3 Divegence The Physical Concept Find the flux going outwad though a sphee of adius. x ρ v0 z a y ψ = D nˆ d = D ˆ

More information

c n ψ n (r)e ient/ h (2) where E n = 1 mc 2 α 2 Z 2 ψ(r) = c n ψ n (r) = c n = ψn(r)ψ(r)d 3 x e 2r/a0 1 πa e 3r/a0 r 2 dr c 1 2 = 2 9 /3 6 = 0.

c n ψ n (r)e ient/ h (2) where E n = 1 mc 2 α 2 Z 2 ψ(r) = c n ψ n (r) = c n = ψn(r)ψ(r)d 3 x e 2r/a0 1 πa e 3r/a0 r 2 dr c 1 2 = 2 9 /3 6 = 0. Poblem {a} Fo t : Ψ(, t ψ(e iet/ h ( whee E mc α (α /7 ψ( e /a πa Hee we have used the gound state wavefunction fo Z. Fo t, Ψ(, t can be witten as a supeposition of Z hydogenic wavefunctions ψ n (: Ψ(,

More information

Chapter 2: Basic Physics and Math Supplements

Chapter 2: Basic Physics and Math Supplements Chapte 2: Basic Physics and Math Supplements Decembe 1, 215 1 Supplement 2.1: Centipetal Acceleation This supplement expands on a topic addessed on page 19 of the textbook. Ou task hee is to calculate

More information

Physics 107 TUTORIAL ASSIGNMENT #8

Physics 107 TUTORIAL ASSIGNMENT #8 Physics 07 TUTORIAL ASSIGNMENT #8 Cutnell & Johnson, 7 th edition Chapte 8: Poblems 5,, 3, 39, 76 Chapte 9: Poblems 9, 0, 4, 5, 6 Chapte 8 5 Inteactive Solution 8.5 povides a model fo solving this type

More information

ME 210 Applied Mathematics for Mechanical Engineers

ME 210 Applied Mathematics for Mechanical Engineers Tangent and Ac Length of a Cuve The tangent to a cuve C at a point A on it is defined as the limiting position of the staight line L though A and B, as B appoaches A along the cuve as illustated in the

More information

Look over Chapter 22 sections 1-8 Examples 2, 4, 5, Look over Chapter 16 sections 7-9 examples 6, 7, 8, 9. Things To Know 1/22/2008 PHYS 2212

Look over Chapter 22 sections 1-8 Examples 2, 4, 5, Look over Chapter 16 sections 7-9 examples 6, 7, 8, 9. Things To Know 1/22/2008 PHYS 2212 PHYS 1 Look ove Chapte sections 1-8 xamples, 4, 5, PHYS 111 Look ove Chapte 16 sections 7-9 examples 6, 7, 8, 9 Things To Know 1) What is an lectic field. ) How to calculate the electic field fo a point

More information

Review. Electrostatic. Dr. Ray Kwok SJSU

Review. Electrostatic. Dr. Ray Kwok SJSU Review Electostatic D. Ray Kwok SJSU Paty Balloons Coulomb s Law F e q q k 1 Coulomb foce o electical foce. (vecto) Be caeful on detemining the sign & diection. k 9 10 9 (N m / C ) k 1 4πε o k is the Coulomb

More information

Electromagnetism Physics 15b

Electromagnetism Physics 15b lectomagnetism Physics 15b Lectue #20 Dielectics lectic Dipoles Pucell 10.1 10.6 What We Did Last Time Plane wave solutions of Maxwell s equations = 0 sin(k ωt) B = B 0 sin(k ωt) ω = kc, 0 = B, 0 ˆk =

More information

7.2.1 Basic relations for Torsion of Circular Members

7.2.1 Basic relations for Torsion of Circular Members Section 7. 7. osion In this section, the geomety to be consideed is that of a long slende cicula ba and the load is one which twists the ba. Such poblems ae impotant in the analysis of twisting components,

More information

Right-handed screw dislocation in an isotropic solid

Right-handed screw dislocation in an isotropic solid Dislocation Mechanics Elastic Popeties of Isolated Dislocations Ou study of dislocations to this point has focused on thei geomety and thei ole in accommodating plastic defomation though thei motion. We

More information

Collaborative ASSIGNMENT Assignment 3: Sources of magnetic fields Solution

Collaborative ASSIGNMENT Assignment 3: Sources of magnetic fields Solution Electicity and Magnetism: PHY-04. 11 Novembe, 014 Collaboative ASSIGNMENT Assignment 3: Souces of magnetic fields Solution 1. a A conducto in the shape of a squae loop of edge length l m caies a cuent

More information

Ch 30 - Sources of Magnetic Field! The Biot-Savart Law! = k m. r 2. Example 1! Example 2!

Ch 30 - Sources of Magnetic Field! The Biot-Savart Law! = k m. r 2. Example 1! Example 2! Ch 30 - Souces of Magnetic Field 1.) Example 1 Detemine the magnitude and diection of the magnetic field at the point O in the diagam. (Cuent flows fom top to bottom, adius of cuvatue.) Fo staight segments,

More information

Physics 1502: Lecture 4 Today s Agenda

Physics 1502: Lecture 4 Today s Agenda 1 Physics 1502: Today s genda nnouncements: Lectues posted on: www.phys.uconn.edu/~cote/ HW assignments, solutions etc. Homewok #1: On Mastephysics today: due next Fiday Go to masteingphysics.com and egiste

More information

8.022 (E&M) - Lecture 2

8.022 (E&M) - Lecture 2 8.0 (E&M) - Lectue Topics: Enegy stoed in a system of chages Electic field: concept and poblems Gauss s law and its applications Feedback: Thanks fo the feedback! caed by Pset 0? Almost all of the math

More information

Vectors, Vector Calculus, and Coordinate Systems

Vectors, Vector Calculus, and Coordinate Systems Apil 5, 997 A Quick Intoduction to Vectos, Vecto Calculus, and Coodinate Systems David A. Randall Depatment of Atmospheic Science Coloado State Univesity Fot Collins, Coloado 80523. Scalas and vectos Any

More information

Chapter 25. Electric Potential

Chapter 25. Electric Potential Chapte 25 Electic Potential C H P T E R O U T L I N E 251 Potential Diffeence and Electic Potential 252 Potential Diffeences in a Unifom Electic Field 253 Electic Potential and Potential Enegy Due to Point

More information

Sections and Chapter 10

Sections and Chapter 10 Cicula and Rotational Motion Sections 5.-5.5 and Chapte 10 Basic Definitions Unifom Cicula Motion Unifom cicula motion efes to the motion of a paticle in a cicula path at constant speed. The instantaneous

More information

Newton s Laws, Kepler s Laws, and Planetary Orbits

Newton s Laws, Kepler s Laws, and Planetary Orbits Newton s Laws, Keple s Laws, and Planetay Obits PROBLEM SET 4 DUE TUESDAY AT START OF LECTURE 28 Septembe 2017 ASTRONOMY 111 FALL 2017 1 Newton s & Keple s laws and planetay obits Unifom cicula motion

More information

Section 8.2 Polar Coordinates

Section 8.2 Polar Coordinates Section 8. Pola Coodinates 467 Section 8. Pola Coodinates The coodinate system we ae most familia with is called the Catesian coodinate system, a ectangula plane divided into fou quadants by the hoizontal

More information

Anyone who can contemplate quantum mechanics without getting dizzy hasn t understood it. --Niels Bohr. Lecture 17, p 1

Anyone who can contemplate quantum mechanics without getting dizzy hasn t understood it. --Niels Bohr. Lecture 17, p 1 Anyone who can contemplate quantum mechanics without getting dizzy hasn t undestood it. --Niels Boh Lectue 17, p 1 Special (Optional) Lectue Quantum Infomation One of the most moden applications of QM

More information

17.1 Electric Potential Energy. Equipotential Lines. PE = energy associated with an arrangement of objects that exert forces on each other

17.1 Electric Potential Energy. Equipotential Lines. PE = energy associated with an arrangement of objects that exert forces on each other Electic Potential Enegy, PE Units: Joules Electic Potential, Units: olts 17.1 Electic Potential Enegy Electic foce is a consevative foce and so we can assign an electic potential enegy (PE) to the system

More information

( ) [ ] [ ] [ ] δf φ = F φ+δφ F. xdx.

( ) [ ] [ ] [ ] δf φ = F φ+δφ F. xdx. 9. LAGRANGIAN OF THE ELECTROMAGNETIC FIELD In the pevious section the Lagangian and Hamiltonian of an ensemble of point paticles was developed. This appoach is based on a qt. This discete fomulation can

More information

Homework # 3 Solution Key

Homework # 3 Solution Key PHYSICS 631: Geneal Relativity Homewok # 3 Solution Key 1. You e on you hono not to do this one by hand. I ealize you can use a compute o simply look it up. Please don t. In a flat space, the metic in

More information

Math Notes on Kepler s first law 1. r(t) kp(t)

Math Notes on Kepler s first law 1. r(t) kp(t) Math 7 - Notes on Keple s fist law Planetay motion and Keple s Laws We conside the motion of a single planet about the sun; fo simplicity, we assign coodinates in R 3 so that the position of the sun is

More information

3.6 Applied Optimization

3.6 Applied Optimization .6 Applied Optimization Section.6 Notes Page In this section we will be looking at wod poblems whee it asks us to maimize o minimize something. Fo all the poblems in this section you will be taking the

More information

Phys102 Second Major-182 Zero Version Monday, March 25, 2019 Page: 1

Phys102 Second Major-182 Zero Version Monday, March 25, 2019 Page: 1 Monday, Mach 5, 019 Page: 1 Q1. Figue 1 shows thee pais of identical conducting sphees that ae to be touched togethe and then sepaated. The initial chages on them befoe the touch ae indicated. Rank the

More information

Potential Energy. The change U in the potential energy. is defined to equal to the negative of the work. done by a conservative force

Potential Energy. The change U in the potential energy. is defined to equal to the negative of the work. done by a conservative force Potential negy The change U in the potential enegy is defined to equal to the negative of the wok done by a consevative foce duing the shift fom an initial to a final state. U = U U = W F c = F c d Potential

More information

Lecture 7: Angular Momentum, Hydrogen Atom

Lecture 7: Angular Momentum, Hydrogen Atom Lectue 7: Angula Momentum, Hydogen Atom Vecto Quantization of Angula Momentum and Nomalization of 3D Rigid Roto wavefunctions Conside l, so L 2 2 2. Thus, we have L 2. Thee ae thee possibilities fo L z

More information

DOING PHYSICS WITH MATLAB COMPUTATIONAL OPTICS

DOING PHYSICS WITH MATLAB COMPUTATIONAL OPTICS DOING PHYIC WITH MTLB COMPUTTIONL OPTIC FOUNDTION OF CLR DIFFRCTION THEORY Ian Coope chool of Physics, Univesity of ydney ian.coope@sydney.edu.au DOWNLOD DIRECTORY FOR MTLB CRIPT View document: Numeical

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department. Problem Set 10 Solutions. r s

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department. Problem Set 10 Solutions. r s MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Depatment Physics 8.033 Decembe 5, 003 Poblem Set 10 Solutions Poblem 1 M s y x test paticle The figue above depicts the geomety of the poblem. The position

More information

CHAPTER 10 ELECTRIC POTENTIAL AND CAPACITANCE

CHAPTER 10 ELECTRIC POTENTIAL AND CAPACITANCE CHAPTER 0 ELECTRIC POTENTIAL AND CAPACITANCE ELECTRIC POTENTIAL AND CAPACITANCE 7 0. ELECTRIC POTENTIAL ENERGY Conside a chaged paticle of chage in a egion of an electic field E. This filed exets an electic

More information