Electic Potentil nd Euipotentils U
Electicl Potentil Review: W wok done y foce in going fom to long pth. l d E dl F W dl F θ Δ l d E W U U U Δ Δ l d E W U U U U potentil enegy electic potentil Potentil diffeence is the wok done pe unit chge y the electic field s the chge moves fom to. Only chnges in e impotnt; cn choose zeo t ny point. Let t infinity nd, then: l d E llows us to clculte eveywhee if we know E
Potentil fom chged spheicl shell E-field (fom Guss' Lw) < R: >R: E E 2 R R R Potentil R > R: E dl E d < R: E dl R E d E d R E d Outside: sme s point chge t cente! R
ELECTRIC POTENTIAL fo Chged Spheicl Conducto Suppose we hve chged Metl sphee with chge. Wht is the electic potentil s function of dius? The potentil is constnt inside conducto.
Lectue, ACT A point chge is fixed t the cente of n unchged conducting spheicl shell of inne dius nd oute dius. Wht is the vlue of the potentil t the inne sufce of the spheicl shell? () () 4 πε (c)
Lectue, ACT A point chge is fixed t the cente of n unchged conducting spheicl shell of inne dius nd oute dius. Wht is the vlue of the potentil t the inne sufce of the spheicl shell? E out () () (c) How to stt?? The only thing we know out the potentil is its definition: E dl To clculte, we need to know the electic field E Outside the spheicl shell: ˆ Apply Guss Lw to sphee: E 2 Inside the spheicl shell: E E d l E d l
Peflight 6: Two spheicl conductos e septed y lge distnce. They ech cy the sme positive chge. Conducto A hs lge dius thn conducto B. A B 2) Compe the potentil t the sufce of conducto A with the potentil t the sufce of conducto B. ) A > B ) A B c) A < B
Electicl Potentil Two wys to find t ny point in spce: Use electic field: E dl Sum o Integte ove chges: 2 2 3 P i i i 3 Fo chge distiution: d P tet like E fom chge distiution d ut scl ddition Exmples of integting ove distiution of chge: line of chge see next slides ing of chge see next slides disk of chge Be le to do these.
Continuous Chge Distiutions Fun with clculus. Exmple: on the xis of line of chge. t P? x d P d d L dx d λ dx ( x x) x x x x); ( d λ L dx ( x x) λ dx Integte using sustitution of viles: u x x; du -dx λ πε ln ( x x 4 L )
Continuous Chge Distiutions Exmple: on xis of ing of chge. t P? d d 4 πε d x d 2 2 z d y 2 2 x P x x d d x x 2 2 2 2 4 πε x 2 2 Result fo x? Result fo x >>?
Potentil fom chged sphee () ( ) (whee ) E Euipotentil The electic field of the chged sphee hs spheicl symmety. The potentil depends only on the distnce fom the cente of the sphee, s is expected fom spheicl symmety. Theefoe, the potentil is constnt on sphee which is concentic with the chged sphee. These sufces e clled euipotentils. Notice tht the electic field is pependicul to the euipotentil sufce t ll points.
Euipotentils Defined s: The locus of points with the sme potentil. Exmple: fo point chge, the euipotentils e sphees centeed on the chge. Why?? Along the sufce, thee is NO chnge in (it s n euipotentil!) Theefoe, A B We cn conclude then, tht The electic field is lwys pependicul to n euipotentil sufce! B A E dl A E dl B Δ E dl is zeo. If the dot poduct of the field vecto nd the displcement vecto is zeo, then these two vectos e pependicul, o the electic field is lwys pependicul to the euipotentil sufce.
EXAMPLES of Euipotentil Lines
Conductos Clim The sufce of conducto is lwys n euipotentil sufce (in fct, the entie conducto is n euipotentil). Why?? If sufce wee not euipotentil, thee would e n electic field component pllel to the sufce nd the chges would move!!
Peflight : A B 3) The two conductos e now connected y wie. How do the potentils t the conducto sufces compe now? ) A > B ) A B c) A < B 4) Wht hppens to the chge on conducto A fte it is connected to conducto B? ) A inceses ) A deceses c) A doesn t chnge
Chge on Conductos? How is chge distiuted on the sufce of conducto? KEY: Must poduce E inside the conducto nd E noml to the sufce. Spheicl exmple (with little off-cente chge): - - - - - - - - - - - - - - - E inside conducting shell. chge density induced on inne sufce non-unifom. chge density induced on oute sufce unifom E outside hs spheicl symmety centeed on spheicl conducting shell.
Euipotentil Exmple Field lines moe closely spced ne end with most cuvtue highe E-field Field lines to sufce ne the sufce (since sufce is euipotentil). Ne the sufce, euipotentils hve simil shpe s sufce. Euipotentils will look moe cicul (spheicl) t lge.