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1 of 2 PHYS3320 Fall 2009 Phys 3320, HW #11, Due at start of class, Wednesday December 2nd Q1. MAXWELL’S EQUATIONS IN GENERAL POTENTIAL FORM Now that we have the full Maxwell’s equations for E and B that describe static and non-static fields, we can see how Maxwell’s equations look in terms of the scalar and vector potential. A. While we still have B = A, in general, E  V for non-static fields. Explain why the formula for B in terms of A is unchanged, while that from E needs an additional term. Show that the additional term is A /t. B. Now substitute these relations for E and B in terms of V and A into Maxwell’s equations and show that the general equations for the potentials V r,t() and Ar,t() (without using any particular gauge condition) are 2V +tA()=  0 2A μ002At2  A +μ00Vt      = μ0J C. The equations in a particular gauge can be obtained from the general equations by substituting in the gauge condition. Derive the differential equations for V and A in the Lorentz gauge and in the Coulomb gauge by this method. In the Lorentz gauge A +μ0 0Vt= 0, whereas in the Coulomb gauge A = 0. D. Griffiths proves that the general solutions to these equations in the Lorentz gauge are V (r,t) =140(r', tr)r r'd',A(r,t) =μ04J(r', tr)r r'd', which involve the retarded time tr= t r  r'c; for this reason these are called the retarded potentials and satisfy our sense of causality. Prove that the advanced potentials in which tr is replace with the advanced time ta= t +r  r'c are also solutions to Maxwell equations in potential form, in the Lorentz gauge. You are proving in fact that the laws of electrodynamics are invariant under time reversal! Despite Griffiths’ dismissive comments about these potentials, they have enormous impact in quantum field theory where anti-matter particles follow a time dependence that is mathematically equivalent to moving backward in time.2 of 2 PHYS3320 Fall 2009 Q2. TIME DEPENDENT FIELDS FROM AN INFINITE WIRE Consider an infinite straight wire in which the current I = kt (k is a constant) begins to flow at t = 0; there is no current for t < 0. The wire remains neutral for all time, and we will magically assume that the current begins to flow at t = 0 everywhere along the entire length of the wire(!). A. Using the retarded potentials in the Lorentz gauge (Griffiths’ Eq. 10.19, p. 423), find the scalar and vector potentials (V, A) as a function of time in all space. B. Using your solutions to part A, find the E and B fields in all space. Compare your results to what you would have found in the quasi-static approximation (see Example 7.9, Griffiths, p. 308). Q3. RADIATION FROM A FALLING POINT CHARGE An electron is released from rest and falls under the influence of gravity. In previous mechanics courses we have probably completely ignored the fact that this charge is losing energy by radiation as it accelerates down! So here we will investigate how large an error we have been making; of all the charged particles in nature, the effect will be largest for an electron, so we’ll take that as our worst case. First, assume that the effect is negligible so that you can simply find the position as a function of time, then calculate the power radiated in the first centimeter of falling. What fraction of the change in gravitational potential energy is this? Is this radiation effect negligible? If the effect is large, we need to go back and include the radiation reaction force in Newton’s 2nd law, in order to find the correct trajectory. Q4. EM PLANE WAVES IN POTENTIAL FORM Find the E and B fields for the scalar and vector potentials V (r,t) = 0, A(r,t) = A0ˆ y sin(kx t), where A0, , and k are constants. Now check that they satisfy Maxwell’s equations, especially noting any constraints required for the constants A0, , and k. In general, now you can see why in quantum mechanics, where we cast the theory in terms of a Hamiltonian (energy function) as opposed to forces, the vector potential A plays the major role when we deal with EM


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CU-Boulder PHYS 3320 - HW #11

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