Physics 1, Fall 1 7 Septembe 1 Today in Physics 1: getting V fom E When it s best to get V fom E, athe than vice vesa V within continuous chage distibutions Potential enegy of continuous chage distibutions Capacitance Potential enegy in capacitos Paallel-plate and cylindical capacitos 84-faad supecapacito module (ICD) 7 Septembe 1 Physics 1, Fall 1 1 When it s easie to get V fom E Even if a distibution does not stetch to infinity, thee ae cases in which one cannot use the naïve Coulomb s law fom of the potential, V kdq -. Tivial example: a chaged conducting sphee. Chage unifom on suface, so V = kq/ thee. But E = inside, so the suface and Q inteio ae an equipotential: b V Ed b a so V = kq/ eveywhee inside. b a Yet kdq -. 7 Septembe 1 Physics 1, Fall 1 Example 1: V within a unifom chaged sphee So emembe to include potential fom oute chage, when computing Vwithina distibution of chage. Usually this is easiest to do by using the foolpoof expession fo potential: V Ed Example: the unifom sphee of chage, fo which we calculated E using Gauss s law (18 Sept 1): 4kˆ 3, E 3ˆ 4k 3, 7 Septembe 1 Physics 1, Fall 1 3 (c) Univesity of ocheste 1
Physics 1, Fall 1 7 Septembe 1 V within a unifom chaged sphee (continued) Because E changes fom at =, we have to split the integal into a sum of two: 3 4k d 4k VEd dd 3 3 3 4 k 1 4 k 3 3 4k 4k. 3 6 kdq kq 4k 4k Compae d ( V!). - 3 7 Septembe 1 Physics 1, Fall 1 4 Example : V within a non-unifom chaged sphee Conside a sphee with chage density given by 1,, Calculate the total chage g Q, Q and V,. Fist find E. Since is spheically symmetic we get to use Gauss s Law with a spheical Gaussian suface: EdA E 4 4kQencl E kqencl 7 Septembe 1 Physics 1, Fall 1 5 V within a non-unifom chaged sphee (cont d) Qencl dv 1 4 d 1 4 4 d d 3 5 4 3 5 While we e at it, get the total chage: 3 5 3 8 Q 4 3 15 5 7 Septembe 1 Physics 1, Fall 1 6 (c) Univesity of ocheste
Physics 1, Fall 1 7 Septembe 1 V within a non-unifom chaged sphee (cont d) So outside the sphee ( > ), E ˆ kq, while inside 3 3 it s E 15kQ 4k ˆ ˆ 3 3 3 5 5 Now we e eady to calculate V V. Again A i the integal i t l beaks b k into two: V Ed 3 d 15kQ kq d 3 3 5 7 Septembe 1 Physics 1, Fall 1 7 V within a non-unifom chaged sphee (cont d) 4 1 15kQ V kq 3 6 4 4 kq 15kQ 3 6 6 kq kq 4 4 5 3 1 7 8 Check: kq kq 4 4 4 V 5 3 1 7 8 kq 7 Septembe 1 Physics 1, Fall 1 8 Electostatic potential enegy of a continuous distibution of chage Similaly, one must be caeful about using U = qv to calculate electostatic potential enegy of continuous distibutions of chage. Hee s an example (poblem 3-64 in the book). A sphee with adius contains a total chage Q, unifomly distibuted though its volume. Calculate its electostatic potential enegy. Imagine building the sphee by binging d infinitesimal shells of the ight chage density in fom infinity one by one. 7 Septembe 1 Physics 1, Fall 1 9 (c) Univesity of ocheste 3
Physics 1, Fall 1 7 Septembe 1 Electostatic potential enegy of a continuous distibution of chage (continued) Since the chage is unifomly distibuted, the density is Q 3Q v 3 4 and if the infinitesimal chage bought fom infinity is spead d thick at this density, onto a sphee of adius composed of the shells which aived peviously, its value can be witten as dq 4 d dq d 7 Septembe 1 Physics 1, Fall 1 1 Electostatic potential enegy of a continuous distibution of chage (continued) Binging the shell in fom infinity involves wok, and this wok is the potential enegy of the shell-peviouslyexisting sphee combination: du dw V dq Since thee s no chage outside by Gauss s adius befoe the new shell Law aives, the potential at is calculated fom q d kq VEd kq d k dq 4 d 4k 3 7 Septembe 1 Physics 1, Fall 1 11 Electostatic potential enegy of a continuous distibution of chage (continued) Combine the last two esults and integate: 4 du V dq k 4 d 3 5 4 4 4 U k d k 3 3 5 3 4 3 3 kq k 3 5 5 Compae to the gavitational potential enegy of a unifom mass 3 GM M (PHY 11): U. 5 7 Septembe 1 Physics 1, Fall 1 1 dq d (c) Univesity of ocheste 4
Physics 1, Fall 1 7 Septembe 1 Capacitance Conside two conductos, chaged up to Q and Q. They ae equipotentials, and the voltage between them is V Ed, and nea each suface the electic field magnitude is E 4 k (18 Sept 1). Suppose we double the value of. What happens to the othe quantities? E Q -Q 7 Septembe 1 Physics 1, Fall 1 13 Capacitance (continued) Doubling doubles the total chage. It also doubles the magnitude of the electic field, but not the patten of field lines (just daw moe of them). And since it doubles the field, it doubles the voltage between the conductos. Appaently, dq is popotional to dv, so Q and V ae popotional. We call the popotionality facto the capacitance, C: Q = CV. E Q -Q 7 Septembe 1 Physics 1, Fall 1 14 What good is C? Enegy stoage in capacitos Capacitos ae impotant as electic cicuit elements. Cicuits can stoe enegy in, and eclaim enegy fom, capacitos. Conside, fo instance, caying a chage Q fom one conducto to the othe, one infinitesimal chage dq at a time: q dw Vdq dq C Q 1 1 Q 1 U W qdq C V. C C Usually we speak loosely about potential V and potential diffeence V in cicuits, and often wite Q CV o U CV. 7 Septembe 1 Physics 1, Fall 1 15 (c) Univesity of ocheste 5
Physics 1, Fall 1 7 Septembe 1 Calculation of the capacitance of aangements of conductos Othe mateials besides conductos have capacitance, but aangements of conductos lend themselves to staightfowad calculation of C. Usually this goes as follows: Pesume electic chage to be pesent; say, Q if thee is only one conducto, o ±Q if thee ae two. Eithe: Calculate the electic field fom the chages, and integate it to find the potential diffeence V between the conductos, o Solve fo the potential diffeence diectly, using V kdq -. Then C = Q/V. 7 Septembe 1 Physics 1, Fall 1 16 Paallel-plate capacito Conside two paallel conducting plates, sepaated by a distance d that is vey small compaed to thei extent in othe dimensions. Suppose each plate has aea A. It doesn t matte what the shape of the flat plates ae, as long as they ae paallel and vey close togethe. With chages Q on the plates, the chage densities ae unifom and have values Q A. w w w A d A d d A 7 Septembe 1 Physics 1, Fall 1 17 Paallel-plate capacito (continued) At points well inside the gap, the plates can be egaded as infinite, to good appoximation. As we found on 11 Septembe 1, the electic field between two oppositely-chaged infinite paallel plates is unifom, with magnitude E 4 k. Q A. d E 4 k Q A. 7 Septembe 1 Physics 1, Fall 1 18 (c) Univesity of ocheste 6
Physics 1, Fall 1 7 Septembe 1 Paallel-plate capacito (continued) So 4 kd Q V Ed 4kd Q A C A A C 4 kd d Q A. d E 4 k Q A. 7 Septembe 1 Physics 1, Fall 1 19 Cylindical capacito Conside two, coaxial, conducting cylindes with adii 1 and > 1. Thei length is L and they cay opposite chages Q (chage pe unit length QL). At points well inside the gap, the cylindes can be egaded as infinite, to good appoximation. 1 L 7 Septembe 1 Physics 1, Fall 1 Cylindical capacito (continued) We have shown by use of Gauss s law (18 Septembe 1) that kˆ, 1 E,, 1 fo infinite, oppositely-chaged coaxial cylindes. 1 L 7 Septembe 1 Physics 1, Fall 1 1 (c) Univesity of ocheste 7
Physics 1, Fall 1 7 Septembe 1 Cylindical capacito (continued) d Q Thus Q V Ed k k ln ; L 1 C 1 L L C. kln 1 ln 1 1 L 7 Septembe 1 Physics 1, Fall 1 Units of capacitance In MKS: the unit of capacitance, named in hono of Michael 1 coul Faaday, is the faad : 1F = 1 volt 1-1 - Note that 8.851 coul Nt m 1-1 -1 885 8.85 1 F m 8.85 885 pf m One faad is a Huge capacitance. Those found aound the lab and in cicuits ae usually in the pf- F ange (1-1 -1-6 F). A 1F paallel-plate capacito with d = 5 m (.1 in) has A =.8 km : 1.7 km on a side, if squae..8 cm 7 Septembe 1 Physics 1, Fall 1 3 (c) Univesity of ocheste 8