Lectue 11: Potentil Gdient nd Cpcito Review: Two wys to find t ny point in spce: Sum o Integte ove chges: q 1 1 q 2 2 3 P i 1 q i i dq q 3 P 1 dq xmple of integting ove distiution: line of chge ing of chge disk of chge
Detemine fom : Detemining fom : Δ d lecticl Potentil Fo n infinitesiml step: dl dl xmple: due to spheicl chge distiution. dl dl dl cosθ Cses: θ : d - dl θ 9 o : d θ 18 o : d - dl(-1) dl Cn wite: d dl ( ( x dx + y dy iˆ + x + z y ˆj + dz) z θ dl d kˆ) ( dxiˆ + dy diectionl deivtive d depends on diection mximum chnge fo θ o 18 degees F θ dl ˆj + dzkˆ)
Potentil Gdient Tke step in x diection: (dy dz ) Similly: And: d ( dx + dy + dz) x y x d dx y y, z const. y z x z ( iˆ + ˆj + x y z ˆ ( ˆ i + j + kˆ) x y z z kˆ) x dx gdient opeto Gdient of points in the diection tht inceses the fstest with espect to chnge in x, y, nd z. points in the diection tht deceses the fstest. pependicul to equilpotentil lines. ptil deivtive
Potentil Gdient xmple: chge in unifom field U qy U/q y whee is tken s t y. ( iˆ + x (iˆ + ˆj + kˆ) ˆ j + y z ˆj kˆ) y Given o in some egion of spce, cn find the othe. Cylindicl nd spheicl symmety cses: Fo dil cse nd is distnce fom point (spheicl) o xis (cylindidl): o y q xmple: of point chge: q ( ) q 1 q ( )( ) 2 2
xmple: The electic potentil in egion of spce is given y (x, y, z) A(x 2 3y 2 + z 2 ) whee A is constnt. Deive n expession fo the electic field t ny point in this egion. ( x iˆ + y ˆj + z kˆ){ A( x 2 3y 2 + z 2) )} (2Axiˆ 6Ay ˆj + 2Azkˆ) 2A( xiˆ 3yj ˆ + zkˆ)
UI7PF7: This gph shows the electic potentil t vious points long the x-xis. 2) At which point(s) is the electic field zeo? A B C D
UI7ACT1 1 The electic potentil in egion of spce is given y ( ) 2 3 x 3x x The x-component of the electic field x t x 2 is () x () x > (c) x <
1 UI7ACT1 The electic potentil in egion of spce is given y 2 3 ( x) 3x x The x-component of the electic field x t x 2 is () x () x > (c) x < We know (x) eveywhee To otin x eveywhee, use x x x x 6x + 3x ( 2 ) 12 + 12 2
CAPACITOR A cpcito is device fomed with two o moe septed conductos tht stoe chge nd electic enegy. Conside ny two conductos nd we put + on nd on. Conducto hs constnt nd conducto hs constnt, then dl Y&F fig. 24.1 The electic field is popotionl to the chges ±. If we doule the chges ±, the electic field doules. Then the voltge diffeence is - popotionl to the chge. This popotionlity depends on size, shpe nd seption of the conductos. const ( )
CAPACITOR, continued If we cll this constnt, Cpcitnce, C, nd the voltge diffeence, -, then, C O C Cpcitnce, depends on the geomety of the two conductos (size, shpe, seption) nd cpcitnce is lwys positive quntity y its definition (voltge diffeence nd chge of + conducto) UNITs of cpcitnce, Coulom/olts o Fds, fte Michel Fdy
xmple: Pllel Plte Cpcito (idel) Clculte the cpcitnce. We ssume +σ, -σ chge densities on ech plte with potentil diffeence : C d A + + + + ----- Need : σa Need : fom def n: Use Guss Lw to find dl
Recll: Two Infinite Sheets (into sceen) Field outside the sheets is zeo Gussin sufce encloses zeo net chge Field inside sheets is not zeo: Gussin sufce encloses non-zeo net chge σa ds A inside σ ε σ σ A A -
xmple: Pllel Plte Cpcito Clculte the cpcitnce: Assume +, - on pltes with potentil diffeence. σ ε Aε d A + + + + dl ----- dl d A ε d Aε d C As expected, the cpcitnce of this cpcito depends only on its geomety (A,d). Note tht C ~ length; this will lwys e the cse!
Cylindicl Cpcito xmple Clculte the cpcitnce: Assume +, - on sufce of cylindes with potentil diffeence. Gussin sufce is cylinde of dius ( < < ) nd length L - + L Apply Guss' Lw: ds 2πL ε 2πε L If we ssume tht inne cylinde hs +, then the potentil is positive if we tke the zeo of potentil to e defined t : dl d d πε L 2πε L 2 ln C 2 πε ln L
Spheicl Cpcito xmple Suppose we hve 2 concentic spheicl shells of dii nd nd chges + nd. uestion: Wht is the cpcitnce? etween shells is sme s point chge +. (Guss s Lw): + - 1 2 o dl dl C d 1 1 ( o ) d 2 d C 1 ( 1 )
Summy A Cpcito is n oject with two sptilly septed conducting sufces. The definition of the cpcitnce of such n oject is: C The cpcitnce depends on the geomety : d A + + + + ----- + - L - + Pllel Pltes Aε d C C Cylindicl 2 πε L ln C Spheicl