PHYS 2421 Fields and Waves. Instructor: Jorge A. López Office: PSCI 209 A, Phone: Textbook: University Physics 11e, Young and Freedman

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1 PHYS 41 Fields and Waves Instructor: Jorge A. López Office: PSCI 9 A, Phone: Textbook: University Physics 11e, Young and Freedman

2 Chapter : Gauss law.1 Charge and electric flux. Calculating electric flux.3 Gauss law.4 Applications of Gauss law.5 Charges on conductors E da 1 Closed Electric Charge Surface flux Enclosed

3 Chapter : An introduction Gauss law: an easy way to calculate electric fields in cases of nice symmetry Consider the charge shown: Place an imaginary sphere around it Calculate the field on the surface: E= k r Calculate the area of the surface A=4 r Calculate the product field x area 1 E A= 4 r 4 r A constant!

4 Chapter : An introduction Notice this result is independent of the radius of sphere Consider the first half: 1 4 r1 E1A1= 4 r1 And for the second half: 1 4 r EA= 4 r Thus for the whole closed surface: E1 A1 E A In general: EiAi All surfaces And for an arbitrary surface: All surface Ei Ai E da Closed Surface Gauss law

5 Chapter : An introduction Use of Gauss law For nice geometries the integral can be done and the magnitude of the field can be extracted Closed Surface E da EA E A Chapter : Gauss law.1 Charge and electric flux. Calculating electric flux.3 Gauss law.4 Applications of Gauss law.5 Charges on conductors

6 Section.1: Charge and electric flux Question: How to determine the charge inside a box without opening? Answer: use a test charge to measure the electric field on the surface The electric field on the surface is related to the electric flux

7 Section.1: Charge and electric flux Examples of electric flux Cases of zero flux Hmwk for Sect..1: 11 th Ed: Question.1 and.5 or 1 th Ed : Question.5 and.3

8 Section.: Calculating electric flux Question: What is the volume of water flowing through the rectangle per unit time? Answer: The flux through it is amount of water that reaches the rectangle in 1 sec. What if the rectangle is tilted? The area gets reduced

9 Section.: Calculating electric flux Now for the electric case Again, the flux is EA For tilted case is EAcos Describing the area with a vector normal to the surface The flux can be represented by a dot product between two vectors E A EAcos

10 Section.: Calculating electric flux And for a closed surface And for a continuous surface All faces E A Surface E da E da E da A Surface E A EAcos3 r Surface m 3 EA cos 3 1 N/C.314 m cos 3 54 Nm /C

11 Section.: Calculating electric flux E Ai E A1 E A6 i1...6 E A EA cos18 EA E A EA cos EA 1 E A EA cos 9 E A E A E A E A E A EA EA E da EdAcos EdA E da EdA E da EA Surface Surface Surface 4 r r As predicted by Gauss law!

12 Section.: Calculating electric flux Summary of Section. Electric flux All faces Surface E A E da Homework for Section.: 11 th Ed: Probls..1 and.5 or 1 th Ed : Probls..1 and.7

13 Section.3: Gauss law Gauss law: When the flux is evaluated over a closed surface, it euals the charge inside the surface divided by o Notice: E da Closed Surface Independent of radius Independent of shape Dependent on charge inside

14 Section.3: Gauss law Find flux on surfaces A, B, C and D A E da Surface B C D Homework for Section.3: 11th Ed: Probls..6, 8 and 11 or 1th Ed : Probls..8, 1 and 13.

15 Section.4: Applications of Gauss law Distribution of charges in a conductor Pick a Add conducting some object charges What will the charges do? What is the field inside the conductor? No field! Repel each other and move as far as possible Use Gauss law with an inside surface A E da E Surface All charges reside on the surface of conductors

16 Section.4: Applications of Gauss law For the field outside, draw a Gaussian surface Use Gauss law: E da EA E4 r E 4 r Surface For the field inside, draw a Gaussian surface Use Gauss law: E da EA E r E Surface 4 Field as a function of r: Homework: 11th Ed: Probl..17 or 1th Ed : Probl..19

17 Section.4: Applications of Gauss law Draw a Gaussian surface Use Gauss law: E da Left Right Cylinder top top On both ends: On cylinder: E da E da E da E da EdAcos9 E da EdAcos EA Cylinder Cyl l l 1 E A rl r l Homework: 11th Ed: Probl..19 or 1th Ed : Probl..1

18 Section.4: Applications of Gauss law Draw a Gaussian surface Use Gauss law: E da Left Right Cylinder top top On cylinder: E da E da E da E da EdAcos9 On tops: A E da E da EdA cos EA Left Right top top top Which agrees with results from infinite disc E

19 Section.4: Applications of Gauss law Draw Gaussian surfaces Use Gauss law: E ; E 1 Add fields in different regions: Left of plates: E E E ˆ ˆ 1 i i E E E Right of plates: ˆ ˆ 1 i i Between plates: E E ˆ ˆ ˆ 1 E i i i

20 Section.4: Applications of Gauss law For the field outside, draw a Gaussian surface Use Gauss law, as before: Q E 4r For the field inside, draw a Gaussian surface Use Gauss law: E da EA E r inside E 4r Charge inside: inside 4 inside Q R r 3 inside Qr 1 Qr 3 3 r r R R E Homework: 11th Ed: Probl.. or 1th Ed : Probl.. 3 Qr R 3 3 Field as a function of r:

21 Section.4: Applications of Gauss law Summary of Section.4 E 1 r E Homework for Section.4: 11 th Ed.: Problem.17,.19,. 1 th Ed.: Problem.19,.1,.

22 Section.5: Charges on conductors Where is the charge in a conductor?

23 Section.5: Charges on conductors Homework for Section.5: 11 th Ed.:.4 1 th Ed.:.6

24 Chapter : Electric Charge and Electric Field.1 Charge and electric flux. Calculating electric flux.3 Gauss law.4 Applications of Gauss law.5 Charges on conductors