Physics 212 Lecture 3, Slide 1 Physics 212 Lecture 3 Today's Concepts: Electric Flux and Field Lines Gauss’s Law

Music Who is the Artist? A)Professor Longhair B)Henry Butler C)David Egan D)Dr. John E)Allen Toussaint WHY? Mardi Gras is 4 weeks from today Physics 212 Lecture 13

Physics 212 Lecture 3, Slide 3 Our Comments SmartPhysics scores will not be visible in the main gradebook for a few weeks You will need to understand integrals in this course!!! Forces and Fields are Vectors Always Draw a Picture First… What do the Forces/Fields Look Like?Prelecture Infinite line of charge E Lecture Finite line of charge (constant ) E Worked Example Finite line of charge (non-constant ) E Homework Arc of charge (constant ) E WORKS FOR ALL !! The “1/4 shell problem” is not special !! It’s the same as all the others!

Physics 212 Lecture 3, Slide 4 Electric Field Lines Direction & Density of Lines represent Direction & Magnitude of E Point Charge: Direction is radial Density  1/R 2 07

Physics 212 Lecture 3, Slide 5 Electric Field Lines Dipole Charge Distribution: Direction & Density much more interesting… simulation 08

Physics 212 Lecture 3, Slide 6 Preflight 3 Checkpoint 3.1 simulation 09 The following three questions pertain to the electric field lines due to the two charges shown. Compare the magnitude of the two charges A. |Q 1 | > |Q 2 |B. |Q 1 | = |Q 2 |C. |Q 1 | < |Q 2 | D. There isn’t enough information to determine the relative magnitude of the charges more field lines emanating from the charge 1, so must be greater in magnitude. “ more field lines emanating from the charge 1, so must be greater in magnitude. ”

Physics 212 Lecture 3, Slide 7 Checkpoint simulation What do we know about the signs of the charges from looking at the picture? A. Q 1 and Q 2 have the same sign B. Q 1 and Q 2 have the opposite sign C. There isn’t enough information to determine the relative signs of the charges “ The field lines leave one and go to the other.. ”

Physics 212 Lecture 3, Slide 8 Preflight 3 Checkpoint Compare the magnitude of the electric field at point A and B. A. |E A | > |E B |B. |E A | = |E B |C. |E A | < |E B | D. There isn’t enough information to determine the relative magnitude of the electric field at points A and B “ Since the electric field lines are denser near point B the magnitude of the electric field is greater at point B than at point A. ”

Physics 212 Lecture 3, Slide 9 Point Charges +2q -q What charges are inside the red circle? +Q -Q +2Q -Q -2Q +Q -Q A B C D E “Telling the difference between positive and negative charges while looking at field lines. Does field line density from a certain charge give information about the sign of the charge?” 13

Physics 212 Lecture 3, Slide 10 Which of the following field line pictures best represents the electric field from two charges that have the same sign but different magnitudes? A B C D simulation 15

Physics 212 Lecture 3, Slide 11 Electric Flux “Counts Field Lines” Flux through surface S Integral of on surface S “I still feel a little unclear about flux integrals. Can you go over some more example problems?” 18

Physics 212 Lecture 3, Slide 12 “ The flux is directly proportional to the charge enclosed by the Gaussian surface, therefore Case 1 will have 2 times the flux of of Case 2 because it has two charges enclosed, versus 1 in Case 2. “ “ The flux is the measure of field lines passing through the surface. Once calculated, it can be shown that the surface area of both cylinders are the same. ” Checkpoint 1(A)  1 =2  2 1=21=21=21=2 (B)  1 =1/2  2 (C) none (D) TAKE s TO BE RADIUS ! L/2 An infinitely long charged rod has uniform charge density  and passes through a cylinder (gray). The cylinder in Case 2 has twice the radius and half the length compared with the cylinder in Case 1.23

Physics 212 Lecture 3, Slide 13 Checkpoint 1(A)  1 =2  2 1=21=21=21=2 (B)  1 =1/2  2 (C) none (D) TAKE s TO BE RADIUS ! L/2 An infinitely long charged rod has uniform charge density l and passes through a cylinder (gray). The cylinder in Case 2 has twice the radius and half the length compared with the cylinder in Case 1. Definition of Flux: E constant on barrel of cylinder E perpendicular to barrel surface (E parallel to dA) Case 1 Case 2 RESULT: GAUSS’ LAW  proportional to charge enclosed ! “ The flux is the measure of field lines passing through the surface. Once calculated, it can be shown that the surface area of both cylinders are the same. ” WHAT IS WRONG WITH THIS REASONING?? THE E FIELD AT THE BARREL SURFACE IS NOT THE SAME IN THE TWO CASES !! 26

Physics 212 Lecture 3, Slide 14 Direction Matters: dAdAdAdA dAdAdAdA dAdAdAdA dAdAdAdA dAdAdAdA E E E E E E E E E For a closed surface, A points outward For a closed surface, A points outward 29

Physics 212 Lecture 3, Slide 15 Direction Matters: dAdAdAdA dAdAdAdA dAdAdAdA dAdAdAdA dAdAdAdA E E E E E E E E E For a closed surface, A points outward For a closed surface, A points outward 30

Physics 212 Lecture 3, Slide 16 Trapezoid in Constant Field Label faces: 1: x = 0 2: z = +a 3: x = +a 4: slanted y z Define  n = Flux through Face nABC  1 < 0  1 = 0  1 > 0 ABC  2 < 0  2 = 0  2 > 0 ABC  3 < 0  3 = 0  3 > 0 ABC   < 0   = 0   > 0 x dA E 31

Physics 212 Lecture 3, Slide 17 Trapezoid in Constant Field + Q Label faces: 1: x = 0 2: z = +a 3: x = +a Define  n = Flux through Face n  = Flux through TrapezoidABC  1 increases  1 decreases  1 remains same Add a charge +Q at (-a,a/2,a/2) +Q How does Flux change?ABC  increases  decreases  remains sameABC  3 increases  3 decreases  3 remains same x y z x

Physics 212 Lecture 3, Slide 18 Q Gauss Law dAdAdAdA dAdAdAdA dAdAdAdA dAdAdAdA dAdAdAdA E E E E E E E E E 41

Physics 212 Lecture 3, Slide 19 Checkpoint 2.3 (A)  increases (B)  decreases (C)  stays same 43 “ The flux throughout the entire surface does not change as the charge moves, for it is still in the sphere. However, if it was moved outside of the sphere, the fluxes would be different because one would be zero! “

Physics 212 Lecture 3, Slide 20 “ charge is closer, field is larger, more lines through a so increase in flux. “ “ as long as the charge is within the shell, the flux will remain the same ” Checkpoint 2.1 (A) d  A increases d  B decreases (B) d  A decreases d  B increases (C) d  A stays same d  B stays same 44

Physics 212 Lecture 3, Slide 21 The total flux is the same in both cases (just the total number of lines) The flux through the right (left) hemisphere is smaller (bigger) for case Think of it this way: 45

Physics 212 Lecture 3, Slide 22 Things to notice about Gauss’s Law If Q enclosed is the same, the flux has to be the same, which means that the integral must yield the same result for any surface. 47

Physics 212 Lecture 3, Slide 23 Things to notice about Gauss’s Law In cases of high symmetry it may be possible to bring E outside the integral. In these cases we can solve Gauss Law for E So - if we can figure out Q enclosed and the area of the surface A, then we know E ! This is the topic of the next lecture… “ Gausses law and what concepts are most important that we know.” 48

Physics 212 Lecture 3, Slide 24 Final Thoughts…. Keep plugging away –Prelecture 4 and Checkpoint 4 due Thursday –Homework 2 due next Tuesday … 7 questions mostly Gauss’ law – try them today! 50