# electrodynamics handwritten notes

The point to remark is that in the proof of an existence theorem of an object, one is generally free to use all heuristic devices that allows one to exhibit the explicit form of such an object. Part 1 Part 2. Put differently, causality demands that the unknown potentials must be determined by their sources ρ and J evaluated at the retarded time. For example, we can formulate the following existence theorem [6]: Given the localised sources \rho ({\bf{x}},t) and {\bf{J}}(x,t) satisfying the continuity equation {\rm{\nabla }}\cdot {\bf{J}}+\partial \rho /\partial t=0 there exist the retarded fields {\bf{F}}({\bf{x}},t) and {\bf{G}}({\bf{x}},t) defined by. endobj be formulated as an existence theorem. Lecture Notes on Quantum Field Theory Kevin Zhou kzhou7@gmail.com These notes constitute a year-long course in quantum eld theory. In the first attempt these equations were defined but not derived and in the second attempt they were not inferred. We then apply the D'Alembertian operator to the retarded potentials, obtaining the wave equations they satisfy. With the aim of implementing this pedagogically interesting idea, we develop in this paper the approach of introducing the scalar and vector potentials before the electric and magnetic fields. The functions { \mathcal P } and {\boldsymbol{ \mathcal A }} in (37) satisfy the wave equations. <>/Border[0 0 0]/P 3 0 R>> Existence Theorem. For example, one of these theories arises when the Faraday induction term of Maxwell's equations is eliminated, obtaining the field equations of a Galilean-invariant instantaneous electrodynamics [30, 31]. Using this expression for {F}^{\mu \nu } together with (23) we obtain, We now take the wave operator {\partial }_{\alpha }{\partial }^{\alpha } to (32), use {\partial }_{\mu }{\partial }^{\mu }G={\delta }^{(4)}(x-x^{\prime} ) and integrate over all spacetime, obtaining the wave equation, Our task will be complete if we appropriately specify the components of the four-current {J}^{\mu }, the four-potential {A}^{\mu }, the electromagnetic field {F}^{\mu \nu } and its dual {}^{* }{F}^{\mu \nu }. Although Maxwell's equations formally imply the continuity equation, the idea that the latter is a consequence of the former is in a sense questionable. where ρ and J are the localised charge and current densities6 We then assume the existence of certain functions of space and time which are causally produced by these localised charge and current densities. In this kind of procedures one makes use of heuristic arguments to show the existence of a mathematical object by providing a method for creating the object. He said that he would start with the vector and scalar potentials, then everything would be much simpler and more transparent. endobj Let us investigate this possibility. BASIC PHYSICS NOTES FOR IIT JAM PHYSICS . A nice interpretation of equation (2) has been given in [6]: 'Consider an observer at a particular location in space who has a watch that reads a particular time. Please do email me if you find any typos or corrections. Hand written notes for IIT JAM PHYSICS, PHYSICS notes, handwritten notes, Post author: dibash; Post published: July 29, 2019; Post category:! In the search for this alternative presentation of Maxwell's equations in which potentials are introduced before fields, we have been motivated by Feynman's words that [38]: '... there is a pleasure in recognising old things from a new point of view. Volume 41, By relativity (to) {\square }^{2}A=j? Corollary 2. End of the story. Classical Electrodynamics: by Eric Poisson, Guelph. A generalised Helmholtz theorem [22, 24] states that an antisymmetric tensor field is completely determined by specifying its divergence and the divergence of its dual. Although the attempt of De Luca et al [3] to make useful Feynman's alternate way to handle electrodynamics is valuable, it turns out to be incomplete because the inhomogeneous Maxwell equations: {\rm{\nabla }}\cdot {\bf{E}}=\rho /{\epsilon }_{0} and {\rm{\nabla }}\times {\bf{B}}={\mu }_{0}{\bf{J}}+{\epsilon }_{0}{\mu }_{0}{\rm{\partial }}{\bf{E}}/{\rm{\partial }}t were not inferred. 3 Author to whom any correspondence should be addressed. 2 0 obj The second attempt was at the end of 1963 as may be seen in the Feynman's handwritten notes recently discovered by Gottlieb [2] and discussed by De Luca et al [3]. Regarding this presentation he wrote: 'I could not think of any really unique or different way of doing it—or any way that would be particularly more exciting than the usual way of presenting it. Combining (6) and (8) one infers equations (10) and (11) which are then identified with the inhomogeneous Maxwell's equations whenever the electric and magnetic fields are defined as (12). Nevertheless, the Lagrangian approach leading to the Lorentz force used by the Luca et al [3] was well-known in the 1960s. Differentiating these potentials one obtains their wave equations (8) and (6). Let us give an example to illustrate our point. The electric and magnetic fields expressed in terms of the scalar and vector potentials are then used in the inhomogeneous Maxwell's equations, obtaining explicit retarded forms of these potentials. Corollary 4. There is a certain parallelism between these two attempts: both were unpublished and both fail to obtain the inhomogeneous Maxwell's equations. Figure 1. J. Phys. By assuming appropriate boundary conditions the solutions of these wave equations yield the retarded potentials which are then differentiated to get the corresponding retarded electric and magnetic fields. An equivalent form of the function G which is Lorentz-invariant is given by {D}_{r}{(x,x^{\prime} )={\rm{\Theta }}({x}_{0}-x{{\prime} }_{0})\delta [(x-x^{\prime} )}^{2}]/(2\pi ), where Θ is the theta function. If for example f(R)=R then [{ \mathcal F }]/R=R[{\bf{F}}]. In his recently discovered handwritten notes on An alternate way to handle electrodynamics'' dated on 1963, Richard P. Feynman speculated with the idea of getting the inhomogeneous Maxwell's equations for the electric and magnetic fields from the wave equation for the vector potential. You will only need to do this once. This is the more essential aspect of a constructive approach. The basic physical ingredients of our axiomatic-heuristic procedure to find these equations were charge conservation mathematically represented by the covariant form of the continuity equation and a heuristic handling of this equation involving the retarded Green function of the wave equation. In other words, the same postulates may lead to distinct spacetime theories! Each of the two final chapters examines a selected experimental issue, introducing students to the work involved in actually proving a law or theory. But in this case they should vanish sufficiently rapidly at spatial infinity so that the surface integrals involving these sources vanish at infinity. © 2020 European Physical Society The four-vector {{ \mathcal A }}^{\nu } satisfies the wave equation {\partial }_{\mu }{\partial }^{\mu }{{ \mathcal A }}^{\nu }={{ \mathcal J }}^{\nu }, where {\partial }_{\mu }{\partial }^{\mu }=-{{\rm{\nabla }}}^{2}+(1/{{ \mathcal C }}^{2}){\partial }^{2}/\partial {t}^{2}. 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