Chapter 4 Fields of Stationary Electric Charges: III Capacitance of An isolated Conductor Capacitance btwn TWo Conductors Potential Energy of a Charge Distribution Energy density in an Electric Field Forces on conductors
Chapter 4 Fields of Stationary Electric Charges : III ◼ Capacitance of An Isolated Conductor ◼ Capacitance btwn Two Conductors ◼ Potential Energy of a Charge Distribution ◼ Energy Density in an Electric Field ◼ Forces on Conductors
4.1 Capacitance of an isolated conductor Consider an isolated conductor, either carrying charges or not We know that the potential v on the conductor is always a constant Both experiments and theory show that, as charge is added to it, its potential rises The magnitude of the change in potential is the amount of charge added and depends on the geomet rical configuration of the conductor as well. This fact can be sum marized as C is called the capacitance of the conductor
Remarks (1) The physical meaning of C is the amount of charge needed to rise the potential by a unit(volt). In SI unit, C has the unit farad coulomb I farad U0 (2)Although C has been defined to be Q/V, it actually depends only on the size and shape of the conductor Example 1. isolated spherical conductor of radius R If it has a charge Q on it, then the potential is 4丌∈ 0 so the capacitance is given by C=Q/V=4丌oR
4.2 Capacitance btwn two conductors Note that an isolated conductor certain restrictions (1)In reality, a conductor is always under influence of the environment So it's difficult to isolate a conductor (2 )An isolated conductor has a small C. For instance a conductor of the size of the earth r=64×10°m, C=4丌0R=7×10 4 F Thus, we need capacitors consisting of two conductors
Example 1. Parallel-plate capacitor(see Fig 4-2) Each of the two plates has an area a and the spacing bwtn them is s One carries a charge @, the other carries-Q So the field btwn is E=F=Eoa and the potential difference is V=Es the capacitance is 02 Say, A=(50 M)2, s=0.1mM, then C X 10+ F. It's greater than that of the earth
A a Figure 4-2 Parallel-plate capacitor, The lower end of the small cylindrical figure is situated inside the lower plate where E=0
Example 2. Concentric-spherical-shell capacitor One has a radius ra and carries a charge Q, the other Rb and-Q(Rb> Ra) So the field btwn the shells is e=9g and the potential difference bwtn the shells is E·dl Qdr Q 1 1 Ra t 4丌∈ 4丌∈ Ra R the capacitance is Q4丌∈n Rb v Rb -Ra As Rb ->0, this approaches to the isolated conduc- tor Taking Ra n rb=60M, rb- Ra=0.1mM, then C is as large as that of the earth
Capacitors Connected in Parallel (see Fig 4-3) Two capacitors C1 and C2 carrying charges Q1 and The potential difference is v, the same for both ca factors The total charge is Q=Q1+Q2=C1V+C2V=(C1+C2) o the total capacitance is Conclusion for capacitors ci i= 1, 2,.. connected in ara e total capacitance is
Q Cr C Figure 4-3 The single capacitor C has the same capacitance as the two capacitors C and C, connected in parallel
Capacitors Connected in Series(see Fig 4-4) Two capacitors C1 and C2, each carries a charge Q The potential differences for C1, and for C2, are The total potential difference is V1+V2 Q x C2 Q(+,) and thus the total capacitance is V C1+C Conclusion: for capacitors cy connected in series, the total capacitance is given by