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2 electrical energy & capacitance today & tomorrow first: wrap up Gauss law, then potential capacitors/dielectrics tomorrow rest of the week: circuits/current/resistance NEXT MON: exam I problembased, cumulative more details throughout the week

3 (a) (b) q q S3 S1 S2 S3 S1 S2

4 (a) Q 3 (b) 1 Q 2 E = Q 1 R 1 R 2

5 (a) (b) E

6 (a) (b)

7 High PE E Low PE A! x B 0 xi xf x q qe! x = xf xi

8 B B! x! x qe A q mg A m E g (a) (b)

9 B B! x! x qe A q mg A m E g (a) (b)

10 r (m)

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12 (a) q 2 (b) P r 12 V 1 = k eq 1 r 12 V 2 = k eq 2 r 12 r 12 P q 1 (c) r 12 q 2 PE = k eq 1 q 2 r 12 q 1 = q 2 V 1 = q 1 V 2

13 (c) r 12 q 2 PE = k eq 1 q 2 r 12 q 1 = q 2 V 1 = q 1 V 2 PE = (1 due to 2) = (2 due to 1) PE = (E to bring 1 close to 2) (E to bring 2 close to 1) PE = (E required to build this thing)

14 oh noes, what about three charges? still just pairs.

15 q1 q 2 P E = P E 1&2 P E 2&3 P E 1&3 = P E 2&1 P E 3&2 P E 3&1 = k e q 1q 3 q 2q 3 r 12 r 13 r 23

16 P E = 1 2 = i=1 3 j=1 j=i k e q i q j r i j ke q 2 q 1 r 21 k eq 3 q 1 r 31 k eq 1 q 2 r 12 k eq 3 q 2 = k e q1 q 2 r 12 q 1q 3 r 13 q 2q 3 r 23 k eq 1 q 3 k eq 2 q 3 r 32 r 13 r 23

17 what is the potential energy of the crystal a q

18 1 2 a we just have to sum the energy of all unique pairs of charges. 3 4 q ways of choosing pairs from five charges = 5 2 so how many are there?! = 5 C 2 = 5! 2! (5 2)! = = 10 (1, 2) (1, 3) (1, 4) (1, 5) (2, 3) (2, 3) (2, 5) (3, 4) (3, 5) (4, 5) #, pairing type separation pairs 4, centercorner a (1, 5) (2, 5) (3, 5) (4, 5) 4, adjacent corners a 2 (1, 4) (3, 4) (2, 3) (1, 2) 2, far corner 2a (1, 3) (2, 4)

19 a q P E square = 4 (energy of centercorner pair) 2 (energy of far corner pair) 4 (energy of adjacent corner pair)»»» ke q 2 ke q 2 ke q 2 = a 2a a 2» = k eq 2 a = k eq 2 a h i 7.83 kq2 a

20 it works for more complicated stuff (a) (b) e e 12 pairs 4 pairs 12 pairs 8 pairs b

21 (a) Rocksalt (b) Rutile Na Cl O 2 Ti 4 M = 1.75 M = 4.82

22 Potential & Conductors travel along surface: B A E perpendicular to path everywhere no work done! can move on surface for free electric force is conservative... potential energy depends only on position, not path

23 q lines of equal potential are perpendicular to field lines equal field = constant force; move this way, change E equal potential = equal energy; can stay here, but need KE

24 ) q q contours of const E = constant force can move along these, but change energy contours of const V = constant energy can move along these without changing E

25 ) q q V contours places you can exist with a certain energy

26 equipotential lines? contours of constant V no work to move along them (like gravity) x, y = spatial coordinates potential constant on lines 2d x, y = spatial coordinates z = electric potential 3d

27 V V

28 ) q conductor = mirror for field & potential lines q q

29 Circuit diagram symbol for voltage sources: Batteries: General constant voltage source:

30 xkcd.com

31 q conductor E=0

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34 C Q Q m!y!v W = mg!y W = " Q!V = " Q 2 /C (a) (b)

35 C Q Q!V C!V (a) (b)

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39 (a) (b) (c) E

40 E plates E dipoles