By Francis E. Low
The writer makes use of a distinct process which emphasizes the sphere theoretic facets of gravitation and the robust analogies among gravitation and the opposite components which are studied in physics. The theory-centered textual content starts off with the best experimental proof then proceeds to the corresponding differential equations, theoretical constructs comparable to strength, momentum and pressure and a number of other functions. End-of-chapter difficulties supply scholars with a chance to check their knowing, function an advent to and a evaluation of fabric no longer incorporated within the publication and will be used to enhance examples, extensions and generalizations of the cloth provided.
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Additional info for Classical Field Theory: Electromagnetism and Gravitation
However, yo11 must choose the right two. 19. aplace equation for the potential 4 becomes $4 -_ rls' + $4 ~ = 0. dy The real and imaginary parts of any analytic function satisfy this equation, as we now show. (a) An analytic function of a complex variable w= f(z), has a derivative which is independent of the path by which Az + O . r (h) Show that thc Cauchy-Riemann equations imply that I I and u are harmonic. x With this choice. show that thc lines of force are lines of constant [ I . 20. , three-dimensional cylinders), one looks For a function LV = J ' ( z ) such that the Chapter 1 Problems 41 cylinders are lines of constant u .
For a static situation dpldt = 0 and V j = 0. That is the case we will be considering in this section. 3) j , = u,, E, . 4) or more generally, The tensor (T,,is called the conductivity. 5) so that the syrnmerric part o f a,, must be positive. The antisymmetric part of (T,,does not contribute to tlWldt. An example of an antisymmetric conductivity tensor can be found in the Hall effect. 6) with B, the magnetic field. 4). 8) identical to the equations for a dielectric medium, with the conductivity replacing the dielectric constant and j replacing D.
Is Fzl = -Flz'? (b) Calculate the torque T~~ exerted by p , on p'. Is 712= - - T ~ ~If? so, fine. If not, explain what happened. 33. Prove directly. using a Green's type theorem, that the coefficients of potential arc symmetric: p,, = P , ~ . 34. (a) Find the Green's function for a conducting insulated sphere. (h) Find the Grecn's function for a conducting sphere held at fixcd potential. (c) Verify the symmetry G(r, r , ) = G ( r , , r ) for both cases. 35. Thc simplest clectrostatic variational calculation i s quite complicated.
Classical Field Theory: Electromagnetism and Gravitation by Francis E. Low