http://xkcd.com/567/
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
(a) (b) q q S3 S1 S2 S3 S1 S2
(a) Q 3 (b) 1 Q 2 E = 0 2 3 Q 1 R 1 R 2
(a) (b) E
(a) (b)
High PE E Low PE A! x B 0 xi xf x q qe! x = xf xi
B B! x! x qe A q mg A m E g (a) (b)
B B! x! x qe A q mg A m E g (a) (b)
r (m)
(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
(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)
oh noes, what about three charges? still just pairs.
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
P E = 1 2 = 1 2 3 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
what is the potential energy of the crystal a q
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)! = 5 4 3 2 1 2 1 3 2 1 = 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)
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 = 4 2 4 a 2a a 2» 4 1 4 2 = k eq 2 a = k eq 2 a h 5 2 2 i 7.83 kq2 a
it works for more complicated stuff (a) (b) e e 12 pairs 4 pairs 12 pairs 8 pairs b
(a) Rocksalt (b) Rutile Na Cl O 2 Ti 4 M = 1.75 M = 4.82
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
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
) 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
) q q V contours places you can exist with a certain energy
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
V V
) q conductor = mirror for field & potential lines q q
Circuit diagram symbol for voltage sources: Batteries: General constant voltage source:
xkcd.com
q conductor E=0
C Q Q m!y!v W = mg!y W = " Q!V = " Q 2 /C (a) (b)
C Q Q!V C!V (a) (b)
(a) (b) (c) E
E plates E dipoles