Electric Potential. and Equipotentials


 Eunice Mitchell
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1 Electic Potentil nd Euipotentils U
2 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
3 Potentil fom chged spheicl shell Efield (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
4 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.
5 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)
6 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
7 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
8 Electicl Potentil Two wys to find t ny point in spce: Use electic field: E dl Sum o Integte ove chges: 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.
9 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 )
10 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 πε x 2 2 Result fo x? Result fo x >>?
11 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.
12 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.
13 EXAMPLES of Euipotentil Lines
14 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!!
15 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
16 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 offcente chge): E inside conducting shell. chge density induced on inne sufce nonunifom. chge density induced on oute sufce unifom E outside hs spheicl symmety centeed on spheicl conducting shell.
17 Euipotentil Exmple Field lines moe closely spced ne end with most cuvtue highe Efield 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.
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