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 a the amount of charge added and depends on the geomet- rical configuration of the conductor as well. This fact can be summarized 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 collom l farad= 1 volt (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 Q 4T∈0 so the capacitance gIven by C=Q/V=4丌o0R
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=6.4x 10%m C=4m0R=7×10-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 Q, the other carries So the field btwn is e=0= and the potential difference is V=es the capacitance is ∈0 Say, A=(50 M),s=0.1mM, then C N 28 X 10+ F. It's greater than that of the earth
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 le other Bb and -Q. (rb> ra So the field btwn the shells is e 丌e0T and the potential difference bwtn the shells is BbE·dl=Ra4 Qdr Q 1 1 a 2 trEo Ra rb le capacitance is Q4丌0B2aB v Rb- Ra As Rb Is approaches to the isolated conduc- tor Taking Ra 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 actors The total charge is Q=Q1+Q2=C1V+C2V=(C1+C2)V, so the total capacitanceIs (C1+C2 Conclusion for capacitors c; i=1, 2,..connected in arallel, the total capacitance is ∑
Q Q+Q Cr C C Figure 4-3 The single capacitor C has the same capacitance as the two capacitors Ci and Ca connected in parallel
Capacitors Connected in Series(see Fig 4-4) Two capacitors Cl and C2, each carries a charge Q The potential differences for Cl, and for C2, are The total potential difference is V=V1+V2 QQ + + and thus the total capacitance is v C1+C Conclusion: for capacitors C i=1, 2,... connected in series, the total capacitance is given by