Physics 102 Fall 2014 NAME: Discussion Session 14 Maxwell’s Equations Worksheet All of classical electricity and magnetism is embodied in the four Maxwell equations, which we investigate in detail in this worksheet. The general solution of Maxwell’s equations ~ We are very familiar can be written in terms of a scalar and vector potential, V and A. with the scalar potential (the voltage), and now we will learn a bit more about the vector potential. Finally, Maxwell’s equations also predict the existence of the electromagnetic field which travels at precisely the speed of light. Hence, Maxwell’s equations not only explain electricity and magnetism, but also light! 1 Conceptual Questions 1. Write down Maxwell’s equations in both integral and differential form, and also their solutions in terms of the scalar and vector potentials. 2. What is the electromagnetic field, and how is it related to the electric and magnetic fields? 3. Maxwell’s equations predict the continuity equation, tion. ∂ρ ∂t ~ Explain this equa= −∇ · J. 4. Why is Maxwell’s modification of Ampere’s law essential to the existence of electromagnetic waves? 5. The presence of magnetic monopoles would require a modification of Gauss’s law for magnetism. Which other Maxwell equation would need modification? 6. When light travels across a given region, what is it that oscillates? What is it that is transported? 7. When astronomers observe a supernova explosion in a distant galaxy, they see a sudden, simultaneous rise in visible light and other forms of electromagnetic radiation. How is this evidence that the speed of light is independent of frequency? 1 2 Maxwell’s Equations 1. Theorists have speculated about the existence of magnetic monopoles, and several experimental searches for such monopoles have occurred. Suppose magnetic monopoles were found and that the magnetic field at a distance r from a monopole of strength qm is given by µ0 qm B= . 4π r2 Modify the Gauss’s law for magnetism equation to be consistent with such a discovery. 2. Maxwell’s equations in a dielectric resemble those in vacuum, but with 0 replaced by κ0 , where κ is the dielectric constant. (a) What are Maxwell’s equations in a dielectric? (b) What is the speed of electromagnetic waves in a dielectric in terms of c? 3. Show that the electric and magnetic fields transformations ~ → A V → are unchanged if we make the simultaneous ~ + ∇λ A , V − ∂λ ∂t where λ (x, y, z, t) is an arbitrary scalar function. Such a transformation is an example of a gauge transformation. 4. This one is tougher. Using the above, show that you can always choose λ such that (a) V = 0. ~ = 0. (b) ∇ · A ~ = −µ0 0 ∂V . (c) ∇ · A ∂t 5. Rewrite Maxwell’s equations, replacing the electric and magnetic fields with the scalar and vector potential. Use each of the gauge transformations in the previous problem and compare the different forms. 6. Suppose ~ (~r, t) = − 1 q θ (vt − r) rˆ and B ~ (~r, t) = 0, E 4π0 r2 where θ is the step function. Show that these fields satisfy all of Maxwell’s equations, ~ Describe the physical situation that gives rise to these fields. and determine ρ and J. 2 7. Consider an electric wave ~ (~r, t) = E˜0 ei(~k·~r−ωt) n ˆ, E where E˜0 is the (in general, complex) amplitude, ~k is the wave vector, which could also be complex, ω is the angular frequency, and ~n is the direction of polarization. (a) Show that, for a wave of this type, the application of the gradient operator is equiv~ ~ alent to replacing the ∇ operator by i~k, (i.e., show that ∇ei(k·~r−ωt) = i~kei(k·~r−ωt) ). Hint: this is easiest to show using Cartesian coordinates! (b) Suppose that the electric field is polarized in the xˆ direction, and moves along the zˆ direction. Using Faraday’s law (and your results from part (a)), determine the magnetic field associated with the above electric field. (c) Suppose that the wave vector ~k had an imaginary part, such that ~k = ~kRe + i~kIm , where ~kRe and ~kIm are the real and imaginary parts of the wave vector, respectively. Explain the effect of the imaginary part on the wave, and give a physical example of where this occurs. ~ = A0 sin(kx − ωt)ˆ 8. Suppose V = 0 and A y , where A0 , ω, and k are constants. ~ = 0. (a) Show that these potentials satisfy the Coulomb gauge, where ∇ · A ~ and B ~ from these potentials. (b) Find E (c) Check that your results from part (b) satisfy Maxwell’s equations in vacuum, where ρ = J~ ≡ 0. What condition must you impose on ω and k? 9. This one is tough! The exact expressions for the electric and magnetic fields at the origin, due to a point charge, q at position ~r, read 2 r ~ = q E c − v 2 ~u + ~r × (~u × ~a) 3 4π0 (~r · ~u) ~ ~ = 1 rˆ × E, B c where ~u ≡ cˆ r − ~v , and ~a is the acceleration of the charge. (a) Using these results, determine the exact electric and magnetic fields at the center of the orbit of a point charge, for which ~r = −Rˆ r (the minus sign is because we ˆ and ~a = −Rω 2 rˆ. Here R is want the fields at the center of the orbit), ~v = Rω θ, the radius of the orbit, and ω is the angular velocity. (b) Check that you get the correct result for your electric field when v c. (c) Recall that the magnetic field at the center of a current loop of radius R is ~ = µ0 I zˆ, B 2R where I is the current. For a point charge in a periodic orbit then I = q/T , where T is the period. Check that you get this same result for your magnetic field. 3

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