tailieunhanh - Basic Theoretical Physics: A Concise Overview P14

Basic Theoretical Physics: A Concise Overview P14. This concise treatment embraces, in four parts, all the main aspects of theoretical physics (I . Mechanics and Basic Relativity, II. Electrodynamics and Aspects of Optics, III. Non-relativistic Quantum Mechanics, IV. Thermodynamics and Statistical Physics). It summarizes the material that every graduate student, physicist working in industry, or physics teacher should master during his or her degree course. It thus serves both as an excellent revision and preparation tool, and as a convenient reference source, covering the whole of theoretical physics. It may also be successfully employed to deepen its readers’ insight and. | 130 17 Electrostatics and Magnetostatics Proof of the above statements is again based on Gauss s integral theorem according to which with En E n Qi Q2 oEnd2M. dVi dVb However dS -dV1 - dV2 where the change of sign reflects that . the outer normal of V1 is an inner normal of S. But according to Gauss s integral theorem End2A m divEdV. Since divE 0 we have Q1 Q2 0 as stated. The solution of the corresponding Neumann problem . not U but Q is given is also essentially unique for the capacitor. Here the proof of uniqueness is somewhat more subtle . we have used the term essentially above . For a given Qj two different solutions k k 1 2 must both satisfy - o Vfa n d2 A Qj for j 1 2 thus their difference w 2 1 satisfies Vw nd2 A 0 dJ dVj and since w is constant on dVj one even has i wVw d Is nd2 A 0 . Moreover according to Gauss s integral theorem it follows that JJJisV wVw dV 0 . i Vw 2 wV2 w dV 0 . Therefore since V2w 0 we have yyyIs Vw 2dV 0 Electrostatic Fields in Vacuo 131 as a consequence Vw 0 everywhere apart from a set of zero measure . Therefore it is not now the potential that is unique but its essence the gradient . the electric field itself. The capacity of a condenser is defined as the ratio of charge to voltage across the plates or C Q. Elementary calculations contained in most textbooks give the following results For a plate capacitor with plate area F and plate separation d a spherical capacitor with inner radius R and outer radius R AR and for a cylindrical capacitor with length L and inner radius R one obtains respectively F R. R AR L C 0d C 4n 0 A R C 2n 0 ln R R . In the limit AR C R AR C R these results are all identical. Numerical Calculation of Electric Fields The uniqueness theorem for the Dirichlet problem is useful amongst other reasons because it allows one immediately to accept a solution found or guessed by any means available as the only solution. One should not underestimate the practical .