Capacitors and Capacitance P07 -

Capacitors:Store Electric Energy Capacitor: two isolated conductors with equal and opposite charges Q and potential difference  V between them. C Q VC Q V Units: Coulombs/Volt or Farads P07 -

Parallel Plate Capacitor Q   AdQ   AQ   AdQ   A E  0E  ?E  0E  0E  ?E  0 P07 - 3

Parallel Plate Capacitor When you put opposite charges on plates, charges move to the inner surfaces of the plates to get as close as possible to charges of the opposite sign P07 - 4

P Calculating E (Gauss’s Law) 000A0000A0   S in q E  dA E  dA  E   QE   Q E  AE  A    A Gauss Gauss Note:We only “consider” a single sheet! Doesn’t the other sheet matter?

Parallel Plate Capacitor top  bottom V  V   E  dS  EdAE  dS  EdA  0 Q d Q 0 AVdQ 0 AVd C C  C depends only on geometric factors A and d P07 - 6

P Spherical Capacitor Two concentric spherical shells of radii a and b What is E? Gauss’s Law  E is not 0 only for a < r < b, where it looks like a point charge:  E  Q r ˆ 4  r 2 0

Capacitance of Earth For an isolated spherical conductor of radius a: C  4  0 a   8.85  10  12 Fma  6.4  10 6 m P C  7  10  4 F  0.7mF A Farad is REALLY BIG!We usually use pF ( ) or nF (10 -9 )   is roughly 10  10 F/m

1 Farad Capacitor How much charge? Q  C  V   1F  1 2 V   12 C P

PRS Question: Changing C Dimensions P

Energy Stored in Capacitor P

Energy To Charge Capacitor -q 1.Capacitor starts uncharged. 2.Carry +dq from bottom to top. Now top has charge q = +dq, bottom -dq 3.Repeat 4.Finish when top has charge q = +Q, bottom -Q P q

Work Done Charging Capacitor At some point top plate has +q, bottom has –q Potential difference is  V = q / C Work done lifting another dq is dW = dq  V +q P q

Work Done Charging Capacitor So work done to move dq is: dW  dq  V  dq q  1 q dqC Total energy to charge to q = Q:  q dq0 q dq0 1 Q1 Q W   dW  C +q -q 1Q2C21Q2C2  P

Energy Stored in Capacitor C QC Q  V V Since 2 Q 2 12C 2Q 2 12C C VC V Q VQ V U U  Where is the energy stored??? P

Energy Stored in Capacitor Energy stored in the E field! C  o AandV EdC  o AandV Ed d Parallel-plate capacitor: U  1 CV 22U  1 CV 22  1  o A  Ed  2   o E 2d 2 2 ( Ad ) u( Ad ) u (volume)(volume) E E2E2 u  E field energy density  o 2 E P

P Farad Capacitor -Energy How much energy? 1 1F 1 2 V 21 1F 1 2 V 2 2 U1 C VU1 C V 2 2 72 J2 72 J Compare to a small capacitor charged to 3kV:  1  100 µ F  3 kV  2  1  1  1 0  4 F  3  V  2  J U1 C| V|2U1 C| V|2 2 2

PRS Question: Changing C Dimensions Energy Stored P