Electric Flux, Surface Integrals, and Gauss s Law
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1 Electic Flux, Suface Integals, and Gauss s Law If we know the locations of all electic chages, we can calculate the Electic Field at any point in space though: v E v E i all i ˆ i 4πε i ρ ˆ 4πε ( ) dv In pactice, except fo vey simple geometies, this is vey difficult! We will avoid this difficulty by using a vey cleve elationship called Gauss s Law This euies the intoduction of citically impotant concepts:. Flux. Suface Integals
2 Flux is the mount of Stuff going though an ea
3 Flux Depends on the oientation of the ea
4 Electic Flux IMPORTNT: Nothing is eally flowing
5 Total Flux Fom a Point Chage Though a Sphee
6 d suface is the total flux though the entie spheical suface d is the diffeential flux though the infinitesimal aea d d E d IMPORTNT NOTE: & ae SCLRS E d suface suface 4πε d 4πε 4π 4πε suface E d d E & ae VECTORS ˆ d ˆ 4πε ˆ ˆ is a unit vecto so d suface 4π d d ˆ ˆ ˆ ()() cos() is the sufaceaea of a sphee ε
7 . Caefully set up the poblem Stategy fo the evaluation of suface integals d suface is the total flux though the entie spheical suface d is the diffeential flux though the infinitesimal aea d d E d IMPORTNT NOTE: & ae SCLRS E d suface suface 4 πε 4πε 4 π 4 πε d E & d ae VECTORS E ˆ d d ˆ 4 πε ˆ d ˆ ˆ is a unit vecto so ˆ ˆ ()() cos() d suface d suface 4 π is the sufaceaea of a sphee ε
8 . Caefully set up the poblem. Wite down expessions fo the vecto uantities Stategy fo the evaluation of suface integals d suface is the total flux though the entie spheical suface d is the diffeential flux though the infinitesimal aea d d E d IMPORTNT NOTE: & ae SCLRS E d suface suface 4 πε 4πε 4 π 4 πε d E & d ae VECTORS E ˆ d d ˆ 4 πε ˆ d ˆ ˆ is a unit vecto so ˆ ˆ ()() cos() d suface d suface 4 π is the sufaceaea of a sphee ε
9 . Caefully set up the poblem. Wite down expessions fo the vecto uantities 3. Make substitution Stategy fo the evaluation of suface integals d suface is the total flux though the entie spheical suface d is the diffeential flux though the infinitesimal aea d d E d IMPORTNT NOTE: E d & d ae SCLRS E & d ae VECTORS suface E ˆ d d ˆ 4 πε d suface ˆ ˆ 4 πε ˆ is a unit vecto so ˆ ˆ ()() cos() d d 4π suface suface 4πε 4 π 4 πε ε is the sufaceaea of a sphee
10 . Caefully set up the poblem. Wite down expessions fo the vecto uantities 3. Make substitution Stategy fo the evaluation of suface integals 4. Use symmety to simplify d suface is the total flux though the entie spheical suface d is the diffeential flux though the infinitesimal aea d d E d IMPORTNT NOTE: & ae SCLRS E d suface suface 4 πε 4πε 4 π 4 πε d E & d ae VECTORS E ˆ d d ˆ 4 πε ˆ d ˆ ˆ is a unit vecto so ˆ ˆ ()() cos() d suface d suface 4 π is the sufaceaea of a sphee ε
11 Using Symmety Both the Electic Field ND the spheical suface have the same symmety. that is they both look the same following a otation about any axis that goes though the cente of the sphee Because they both have the same symmety, it is possible to easily convet the potentially suface integal to an easy to evaluate simple integal. This use of symmety will be citical when we evaluate suface integals in eal poblems.
12 . Caefully set up the poblem. Wite down expessions fo the vecto uantities 3. Make substitution Stategy fo the evaluation of suface integals 4. Use symmety to simplify 5. Evaluate integal, do algeba d suface is the total flux though the entie spheical suface d is the diffeential flux though the infinitesimal aea d d E d IMPORTNT NOTE: & ae SCLRS E d suface suface 4 πε 4πε 4 π 4 πε d E & d ae VECTORS E ˆ d d ˆ 4 πε ˆ d ˆ ˆ is a unit vecto so ˆ ˆ ()() cos() d suface d suface 4 π is the sufaceaea of a sphee ε
13 Total Flux is Independent of the Radius of the Sphee ε sphee E d Q d 4π 4πε 4πε ε sphee Q Q Electic field deceases as / but ea inceases as
14 Total Flux is Independent of the Shape of the Sphee Electic field deceases as / but aea inceases as ND ea of d inceases with φ But cos φ deceases with φ
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16 Thee is no TOTL Flux fo a Chage OUTSIDE a closed suface Evey Electic Field Line That Entes, Must Leave Somewhee Else!
17 The same aguments hold fo positive and negative chages
18 GUSS S LW E E d Q enclosed ε closed suface The total electic flux though a closed Suface is eual to /ε times the total (net) electic chage inside the suface.
19 Using Gauss s Law to Detemine the Electic Field of a Chaged Plane
20 Using Gauss s Law to Detemine the Electic Field of a Chaged Plane
21 Plane of Chage has suface chage density σ chage unit aea Symmety implies that the field is eveywhee pependicula to the plane E d E d Q enclosed ε E d E d + E d caps cylinde Q enclosed ε E caps d + σ ε E is unifom ove end cap E d ove the cylinde E E σ ε σ ε d Magnitude of E is independent of position
22 Example: Conducting Paallel Plates (See Y&F Example.8) Two lage paallel conducting plates ae given chages of eual magnitude and opposite sign; the chage pe unit aea is +σ fo one and σ fo the othe. Detemine whee the chages ae located and what the fields eveywhee. NOTE: It is impotant to note that the actual electic field will have a complicated shape at the edges of the plates We will odinaily disegad these Finging Fields when we do electic field calculations. Sometimes we assume the plates ae infinite o we assume the size of the plate is much, much bigge than the sepaation.
23 Stategy fo Solution of Gauss s Law Poblems: Use a succession of Gaussian Sufaces. Stat with fields o chages you know. Use this infomation to bootstap you way along to lean about the fields o chages you don t know. +σ -σ Impotant Note: The symmety of the poblem implies that the component of the electic field pependicula to the plates must be zeo. closed suface E d Q enclosed ε
24 Question #: What is the Electic Field Outside the Plates? Obsevation: Because the chage densities ae eual and opposite, the total chage will be zeo if the suface encloses eual aea of both plates Pick Suface # to enclose both plates suface# E d E Contibution fom c is zeo because field must be paallel to suface and dot poduct is zeo outside left a + E outside ight b + ε Total chage inside is zeo because of cancellation of + and chages. +σ -σ c E + E ea a euals aea b outside outside left ight a Suface # b Symmety of system tells us the magnitudes of the the fields outside ae eual. Eoutside Eoutside left ight
25 Question #: What is the Chage on the Outside Suface of the Plates? Obsevation: We know that the field outside the plates is zeo fom pevious step. We know the field inside the conducto is zeo. Pick Suface # to pass though egions whee we know E Contibution fom c is zeo because field must be paallel to suface and dot poduct is zeo +σ -σ suface# E d E outside left a + E inside conducto b + Q enclosed ε E outside + Einside Qenclosed ea a euals aea b left conducto Q enclosed c a b Suface # Conclusion: No chage on outside sufaces of Conductos LL CHRGE IS ON INSIDE SURFCE OF CONDUCTORS!
26 Question #3: What is the Electic Field Between the Plates? Pick Suface #3 to have suface b in the egion between the plates and suface a in a egion of zeo field suface# E d E E inside conducto inside conducto a a + E + E between plates between plates E between plates b b Q ε enclosed σ ε σ ε b Total chage is eual to aea times aeal density +σ -σ a + + b + Suface + #
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34 Using Gauss s Law to detemine the Electic Field fom a line Chage
35 Plane of Chage has suface chage density λ chage unit length Symmety implies that the field is eveywhee pependicula to the line chage E d E d Q enclosed ε E d E d + E d caps cylinde Q enclosed ε + E( ) d cylinde λl ε E is unifom ove cylinde E d ove the end caps E ( ) π l λl ε d E( ) πε λ Fo cylindical geomety E falls of as /
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40 Sample CP Poblem
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42 Chapte Summay
43 Chapte Summay (con t)
44 End of Chapte You ae esponsible fo the mateial coveed in T&F Sections.-.5 You ae expected to: Undestand the following: Electic Flux, Vecto ea, Gauss s Law Be able to pefom suface integals in simple geometies (planes, sphees, cylindes, lines, etc..) Be able to apply Gauss s Law fo simple geometies using sufaces with appopiate symmety. Be able to apply Gauss s Law in cases whee chaged and unchaged conductos and insulatos ae involved. Be able to econstuct the easoning used in examples.3 though. Recommended F&Y Execises chapte :,6,3,5,9
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