1 of 2 PHYS3320 Fall 2009 Phys 3320, HW #7, Due at start of class, Wed Oct. 21st Q1. EM PLANE WAVE – I Let’s examine our most basic 3D wave, the sinusoidal plane wave described by E(r,t) = Eocos(k  r  t +), in which E0 is a constant vector equal to E0z, k is the wave vector ky,  is the angular frequency and  is a constant phase shift. A. Describe in words what this mathematical expression represents physically. You may use sketches, but if you do, they should be well described. In particular, you should describe in which direction the wave is moving and with which speed, the wavelength of the wave, and the period of the wave. In addition, make a sketch of E(x=0,y,z=0,t=0) and E(x=0,y=0,z=0,t) for the case  = 90o being careful to indicate the direction of the field and the scales on the y and t axes. How is the field at x=a, E(x=a,y,z=0,t=0), different from the case at x=0? Why is this called a plane wave (i.e., where is (are) the plane(s))? B. Describe how the direction of the electric field changes in time. If the direction of E always lies in one plane, the wave is said to be linearly polarized. Is this wave linearly polarized? C. Find the associated magnetic field B(r,t) for this plane electric wave. Again, sketch B(x=0,y,z=0,t=0) and B(x=0,y=0,z=0,t) indicating the direction of the field and the scales on the y and t axes. D. Calculate the energy density uEM and Poynting vector S for these fields. Interpret the answers physically (that is, explain if they make sense). E. Calculate the momentum density and stress tensor for these fields. Again, interpret the answers physically. F. Calculate the angular momentum density EM(see Griffiths’ p. 358) about the origin (0,0,0). If you integrated this density over a cube of centered at the origin at one instant in time, would the angular momentum in that cube be zero or non-zero? Q2. EM PLANE WAVE IN COMPLEX NOTATION A. Rewrite the E and B fields from Q1 in complex notation. B. Calculate the energy density uEM and Poynting vector S for these fields in complex notation. The original fields were the real parts of the complex exponentials. Do the real parts of uEM and S agree with the answers to part D of Q1? If not, why not? C. Suppose now that we add two plane waves, E1 and E2, (superposition still works!) to find the total electric field. Let E1(r,t) = E1cos(k  r  t +1) and E2(r,t) = E2cos(k  r  t +2) so in this simple case the waves propagate in the same direction. The amplitudes are E1 = E1z and E2 = E2z. Find ET(r,t) = E1(r,t)+E2(r,t) in a form of ET(r,t) = ETcos(k  r  t +T), giving expressions for the total amplitude and phase shift in terms of those from E1(r,t) and E2(r,t). Using the complex notation will greatly simplify this problem!2 of 2 PHYS3320 Fall 2009 Q3. EM PLANE WAVE – II Let’s examine another basic 3D wave, the sinusoidal plane wave described by E(r,t) = E1cos(k  r t +) + E2cos(k  r t ++ /2), in which E1 is a constant vector equal to E0z, E2 is a constant vector equal to E0x, k is the wave vector ky as before,  is the angular frequency, and  is a constant phase shift. A. As in problem Q2, find an expression for the total E(r,t). B. Describe how the direction and magnitude of the electric field change in time. Is this wave linearly polarized? C. Find the associated magnetic field B(r,t) for this plane electric wave. Describe how the direction and magnitude of the magnetic field change in time. D. Calculate the energy density uEM and Poynting vector S for these fields. Interpret the answers physically. Extra Credit. Calculate the angular momentum density EMabout the origin (0,0,0). Interpret your result