8.022 (E&M) - Lecture 2

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1 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 used in the couse is in it Math eiew: too fast? Will eiew new concepts again befoe using them Pace of ectues: too fast? We hae a lot to coe but please emind me! l Last time Coulomb s law: F = qq ˆ upeposition pinciple: F F Q Q i= N qq i = = i = V i ˆ ρ dv Q ˆ i q q q q N dq Q q i ˆi V ˆi q 5 Q eptembe 8, Lectue

2 Enegy associated with F Coulomb y q Q x How much wok do I hae to do to moe q fom to? eptembe 8, Lectue Wok done to moe chages How much wok do I hae to do to moe q fom to? Qqˆ W = F I ds whee F I = F Coulomb =. q ds Assuming adial path: Qqˆ Qq Qq W ( ˆ ) = F I ds = i d = Does this esult depend on the path chosen? No! You can decompose any path in segments // to the adial diection and segments _ to it. ince the component on the _ is nul l the esult does not change. Q eptembe 8, Lectue 4

3 Coollaies The wok pefomed to moe a chage between P and P same independently of the path chosen W = F ds Path = F ds Path = F ds Path y The wok to moe a chage on a close path is eo: W = F ds = 0 Any is the x In othe wods: the electostatic foce is conseatie! eptembe This 8, will 004allow us to intoduce 8.0 the concept Lectue of potential (next week) 5 P P Enegy of a system of chages How much wok does it take to assemble a cetain configuation of chages? y q q q P WQ ( ) = F ds = 0 no othe chages: F=0 qq W + = F I ds = qq qq q q W + + = W + + W + + W + = + + eptembe 8, Lectue x Enegy stoed by N chages: U i= N j= N = qq i j i = j = ij j i 6

4 The electic field Q: what is the best way of descibing the effect of chages? chage in the Uniese F E chages in the Uniese q qq Q F ˆ q = But: the foce F depends on the test chage q define a quantity that descibes the effect of the chage Q on the suoundings: Electic Field F q Q E = = ˆ Units: dynes/e.s.u q eptembe 8, Lectue 7 Electic field lines Visualie the diection and stength of the Electic Field: Diection: // to E, pointing towads and away fom + Magnitude: the dense the lines, the stonge the field. Faucet ink Demo + - Popeties: Field lines nee coss (if so, that s whee E=0) They ae othogona l to equipotential sufaces (wil l see this late). eptembe 8, Lectue 8 4

Electic field of a ing of chage Poblem: Calculate the electic field ceated by a unifomly chaged ing on its axis pecial case: cente of the ing Geneal case: any point P on the axis P Answes: Cente of

5 Electic field of a ing of chage Poblem: Calculate the electic field ceated by a unifomly chaged ing on its axis pecial case: cente of the ing Geneal case: any point P on the axis P Answes: Cente of the ing: E=0 by symmety Geneal case: Q E = ˆ ( + ) eptembe 8, Lectue 9 Electic field of disk of chage Poblem: Find the electic field ceated by a disk of chages on the axis of the disk P Tick: a disk is the sum of an infinite numbe of infinitely thin concentic ings. And we know E ing (ceatie ecycling is fai game in physics) eptembe 8, Lectue 0 5

6 E of disk of chage (cont.) Electic field of a ing of adius : Q E ing ( ) = ˆ ( + ) P If chage is unifomly spead: dq = σ da = π σ d Electic field ceated by the ing is: σ π d de = ˆ ( + ) Integating on : 0: = = σ π d E = de = ˆ = πσ ˆ 0 0 = = ( + ) + eptembe 8, Lectue pecial case : infinity Fo finite : E = πσ ˆ + What if infinity? E.g. what if >>? ince lim = 0 + E = πσ ˆ P Conclusion: Electic Field ceated by an infinite conductie plane: Diection: pependicula to the plane (+/- ) Magnitude: πσ (constant!) eptembe 8, Lectue 6

7 pecial case : h>> Fo finite : E = πσ ˆ + P What happens when h>>? Physicist s appoach: The disk will look like a point chage with Q= σπ E=Q/ Mathematician's appoach: Calculate fom the peious esult fo >> (Taylo expansion): E = πσ ˆ = πσ ˆ + + Q ~ πσ ˆ ( ) = πσ ˆ = ˆ eptembe 8, Lectue / The concept of flux ˆn Conside the flow of wate in a ie The wate elocity is descibed by Define the loop aea ecto as xy (,, ) x ˆ ˆ x + y y + ˆ ( x,, y ) A Anˆ Immese a squaed wie loop of aea A in the wate (suface ) Q: how much wate will flow though the loop? E.g.: What is the flux of the elocity though the suface? eptembe 8, Lectue 4 7

8 What is the flux of the elocity? It depends on how the loop is oiented w..t. the wate Assuming constant elocity and plane loop:. ) if A Φ = 0;. ) if A Φ = A ;. ) if A = θ Φ = A cos θ = i A. ˆn ˆn ˆn θ Φ = da i Geneal case (definition of flux): eptembe 8, Lectue 5 F.A.Q.: what is the diection of da? Defined unambiguously only fo a d suface: At any point in space, da is pependicula to the suface It points towads the outside of the suface Examples:... ˆn Intuitiely: da is oiented in such a way that if we hae a hose inside the suface the flux though the suface will be positie eptembe 8, Lectue 6 8

9 Flux of Electic Field Definition: Φ E Φ = E i da Example: unifom electic field + flat suface Calculate the flux: Φ= Ed = i A EA i = E A cos θ Intepetation: epesent E using field lines: Φ E is popotional to N field lines that go though the loop θ E NB: this intepetation is alid fo any electic field and/o suface! eptembe 8, Lectue 7 Φ E though closed (d) suface Conside the total flux of E though a cylinde: Φ = Φ +Φ + Φ tot Calculate Φ, Φ,Φ Cylinde axis is // to field lines Φ =0 because E nˆ Φ = Φ but opposite sign since Φ= Ed i A = E A cos θ The total flux though the cylinde is eo! da da E eptembe 8, Lectue 8 9

10 Φ E though closed empty suface Q: Is this a coincidence due to shape/oientation of the cylinde? Clue: Think about intepetation of Φ E : popotional # of field lines though the suface Answe: No: all field lines that get into the suface hae to come out! Conclusion: The electic flux though a closed suface that does not contain chages is eo. eptembe 8, Lectue 9 Φ E though suface containing Q Q: What if the suface contains chages? Clue: Think about intepetation of Φ E : the lines will eithe oiginate in the suface (positie flux) o teminate inside the suface (negatie flux) Faucet ink + - Conclusion: The electic flux though a closed suface that does contain a net chage is non eo. eptembe 8, Lectue 0 0

11 imple example: Φ E of chage at cente of sphee +Q Poblem: Calculate Φ E fo point chage +Q at the cente of a sphee of adius olution:. E. // da eeywhee on the sphee Q Point chage at distance : E = ˆ Q Q Q i Φ = E da = da = da = 4 π = 4 π Q eptembe 8, Lectue Φ E though a geneic suface What if the suface is not spheical? Impossible integal? Use intuition and intepetation of flux! +Q Vesion : Conside the sphee Field lines ae always continuous Φ =Φ =4πQ Vesion : Pucell.0 o next lectue Conclusion: The electic flux Φ though any closed suface containing a net chage Q is popotional to the chage enclosed: Φ= E da i = 4 π Q eptembe 8, Lectue enc Gauss s law

12 Thoughts on Gauss s law Φ= EdA i = 4 πq (Gauss's law in integal fom) Why is Gauss s law so impotant? Because it elates the electic field E with its souces Q encl Gien Q distibution find E ( integal fom) Gien E find Q (diffeential fom, next week) Is Gauss s law always tue? Yes, no matte what E o what, the flux is always = 4πQ Is Gauss s law always useful? No, it s useful only when the poblem has symmeties eptembe 8, Lectue Applications of Gauss s law: Electic field of spheical distibution of chages Poblem: Calculate the electic field (eeywhee in space) due to a spheical distibution of positie chages o adius. (NB: solid sphee with olume chage density ρ) y Appoach # (mathematician) I know the E due to a po nt chage dq: de=dq/ I know how to integate ol e the integal inside and outside the sphee (e.g. < and >) ' = ' = ' = ' = ' = 0 ' = 0 ' = 0 ' = 0 i dq ρ dv ρ ' de = = = d θ d φ sin θd θ d φ ' ' '' Comment: coect but usually heay on math! + x Appoach # (physicist) Why would I ee sole an i ntegal is somebody (Gauss) aleady did it fo me? Just use Gauss s theoem Comment: coect, much much less time consuming! eptembe 8, Lectue 4

13 Applications of Gauss s law: Electic field of spheical distibution of chages Physicist s solution: ) Outside the sphee (>) Apply Gauss on a sphee of adius : Φ= Ei da = 4π Q enclosed + ym m ety: E is constant on and to da. Ei da = E 4π = 4π Q Q Fo >, sphee looks E = like a point chage! ) Inside the sphee (<) Apply Gauss on a sphee of adius : A g ain: Φ= Ei da = 4π Q ; sym m e ty: E is c o n s ta nt o n a n d to da. enclosed 4 Ei da = E 4π ; Q = ρ dv = ρ 4 π E = πρ enc eptembe 8, Lectue 5 Do I get full cedit fo this solution? Did I answe the question completely? No! I was asked to detemine the electic field. The electic field is a ecto magnitude and diection How to get the E diection? Look at the symmety of the poblem: pheical symmety E must point adially + C om plete solution: Q E E = ˆ fo > 4 E = πρ ˆ fo < eptembe 8, Lectue 6

14 Anothe application of Gauss s law: Electic field of spheical shell Poblem: Calculate the electic field (eeywhee in space) due to a positiely chaged spheical shell o adius (suface chage density σ) Physicist s solution:apply Gauss ) Outside the sphee (>) Apply Gauss on a sphee of adius : Φ= E i da = 4 π Q enclosed ym m ety: E is constant on and to da. E i da = E = = σ 4 π 4 π Q encl 4 π ( 4 π ) 4 πσ Q E = ˆ = ˆ sam e as point chage! ) Inside the sphee (<) + NB: spheical symmety E is adial Apply Gauss on a sphee of adius. But sphee is hollow Q enclosed=0 E=0 eptembe 8, Lectue 7 till anothe application of Gauss s law: Electic field of infinite sheet of chage Poblem: Calculate the electic field at a distance fom a positiely chaged infinite plane of suface chage density σ Again apply Gauss Tick #: choose the ight Gaussian suface! Look at the symmety of the poblem Choose a cylinde of aea A and height +/- E Tick #: apply Gauss s theoem Φ tot = Φ side + Φ top + Φ bottom ymmety: E // axis Φ side=0 and Φ top = Φ bottom Φ= E i da = 4 π Q cylinde enclosed E i da = EdA = EA = 4 π ( σ A ) cylinde top E = πσ ˆ eptembe 8, Lectue 8 4

15 Checklist fo soling 8.0 poblems ead the poblem (I am not joking!) i i Look at the symmeties befoe choos ng the best coodinate system Look at the symmeties aga n and find out what cancels what and the diection of the ectos inoled Look fo a way to aoid al complicated integation emembe physicists ae lay: complicated integal you scewed up somewhee o thee is an easie way out! Tun the math cank Wite down the comp ete solut on (magnitudes and diect ons fo all the l l i i diffeent egions) Box the so ution: you gades will loe you! If you encounte expansions: Find you expansion coefficient ( x<< ) and massage the esult until you get something that looks like (+x) N, (-x) N, o ln(+x) o e x Don t stop the expansion too ealy: Taylo expansions ae moe than limits l eptembe 8, Lectue 9 ummay and outlook What hae we leaned so fa: Enegy of a system of chages Concept of electic field E To descibe the effect of chages independently fom the test chage Gauss s theoem in integal fom: Φ=. E da i = 4 π Q encl Useful to deie E fom chage distibution with easy calculations Next time: Deie Gauss s theoem in a moe igoous way ee Pucel l.0 if you cannot wait Gauss s law in diffeential fom with some moe into to ecto calculus Useful to deie chage distibution gien the electic fields Enegy associated with an electic field eptembe 8, Lectue 0 5

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