33ng a) Calculate the time-varying W vector, SE = E3: BS. (0K to derive by hand, but typeset your derivation and at least double-check w/ Mathematica == and FullSimplify) ln[l3]:: - your code here , b) Again, using SliceVectorPlot3D, plot the Mating vector at t=0, I think VectorPoints > 20 and VectorScale -> 0405, at plane x==O looks clear, but feel free to experiment with the visual. In[l41:= \\ your code here + c) Before proceeding: Describe the results Where is the energy flowing? From where is it coming? Be careful to pay attention to spatial positioning ofthe Wing vector field lines relative to circular and linear features in the E-field, and peaks and nulls in the B-field. (Your written answer here) d) Now calculate the divergence ofthe amusing vector and use SliceDensityPlot3D to View it. Inll51:= \\ your code here + e) Use Shown to overlay the slice density plot ofV~ with a vector slice plot of g, and another Show to overlay V3 with E. In[l61;: \\ your code here a f) After examining these overlays, was your previous description of power flow correct? If not, What is different? (your written answer here) 4. Intensity, B and E fields For this problem assume we're working with a 500 mW laser, emitting at A=532 um (bright green), with a beam having a cross-sectional area of 1 mmz. (For simplification, assume the intensity is constant across the beam, and that it may be regarded as a plane wave.) 3) Calculate the intensity (irradiance) , in units ofW/ cmz. In[i7]:: ' your code here + b) Calculate the electric field magnitude, in units of kV/m: ln[l8]:: - your code here , 4;) Calculate the magnetic field magnitude, in units of pTv (Hint: see 5,91 2.56 in Em & Ware) Inll9]:: \\ your code here w WOLFRAM MATHEMATICA STUDENT EDITION Learning Center | Wolfram Community | Help Input 5. Lorentz force, B- and E- fields Calculate velocity of electron in an oscillating dipole. As above, assume for the optical field that 1=532 no. Let's assume an electric dipole of an atom having displacement x=1A. It has to traverse this distance in half the oscillation period, so its peak velocity is approximately In[20]:= (* provided code *) v = UnitConvert [Quantity [1., "Angstroms"] * f / 2, "Meters" / "Seconds" ] Meters .. Quantity: Angstroms and - - are incompatible units WALL Seconds Out[20]= f $Failed a) Calculate the contribution to the Lorentz force from E-field from the previous problem, acting on the electron, in units of femtoNewtons (10-15 Newtons). (If you couldn't get previous problem to work, just assume E=10 kV/m). In[21]:= (* your code here *) LA b) Calculate the contribution to the Lorentz force from B-field from the previous problem, acting on the electron, in units of femtoNewtons (10-15 Newtons). (If you couldn't get previous problem to work, just assume B=100 MT). In[22]:= (* your code here *) c) Compare the electric and magnetic Lorentz forces here. Are we justified in ignoring magnetic effects? (your written answer here) d) Now calculate the electron velocity that would cause the magnetic Lorentz force to be equal to electric Lorentz force. In[23]:= (* your code here *) e) How does this velocity compare with the speed of light? If the E-field could accelerate the electron to these speeds, in which direction would the magnetic field then push the electron? Activate Windows (your written answer here) Go to Settings to activate Windows.WOLFRAM MATHEMATICA STUDENT EDITION Learning Center | Wolfram Community Input 1. Superposition of 3D vector plane waves The following code produces a vector plane wave, propagating along the 2 + y direction. (The RotationMatrix takes the k-vector, rotates it by 90-degrees to produce the field vector polarization). NOTE that the k-vector is perpendicular to the field vectors. In[1]:= (* provided code *) r = {x, y, z} w = 1 k1 = {0, 1, 1} E1 = {0, 1, - 1} Cos [kl.r - wt] vecPlot = SliceVectorPlot3D [ E1 / . {t - 0), x =: 0, {x, -5, 5), {y, 0, 10}, {z, 0, 10}, VectorPoints - 20, VectorScale - 0.07, AxesLabel - Automatic, VectorScaling - "Linear"]; annotations = Graphics3D [ {Thick, Black, Arrow[ { {0, 5, 5), {0, 5, 5} +k1}]}]; Show [ { vecPlot, annotations} ] Out[1]= {x, y, z} Out[2]= 1 Out[3]= {0, 1, 1} Out[4]= {0, Cos [t - y - z] , - Cos [t - y - z] } 10 10 Activate Windows Go to Settings to activate WindowsWOLFRAM MATHEMATICA | STUDENT EDITION Learning Center Wolfram Con Input Out 5 10 . . ...... Out[7]= z 5 -5 0 X a) Define a plane wave E2, propagating in the 2 - y direction, and plot it using SliceVectorPlot3D. In[8]:= (* your code here *) b) Define the sum E3=E1+E2 and again make a SliceVectorPlot3D as above. In[9]:= (* your code here *) c) Describe the interference pattern between these two plane waves. Think about the physics and the forces it would impart to a charge or a dipole. If you were to place a charge (e.g. an electron) or a permanent dipole (e.g. water molecule) in this field, what would happen to it? (Your written answer here) Activate Windows Go to Settings to activate WirWOLFRAM MATHEMATICA |STUDENT EDITION Learning Center | Wolfram Communit Input Out 2 B-field of the above superposition a) Given the field E3=(E1+E2) from 6.1, which of Maxwell's equations would you use to calculate the magnetic field? Write it down here. ( Your answer here ) b) Before doing any further calculations, sketch with pencil/pen and paper the magnetic field you expect, and use your intuition to explain how the corresponding magnetic field should behave. Paste your hand-drawn sketch here. c) Derive the corresponding magnetic field. Let w=1 for convenience. (OK to derive by hand, but typeset your derivation and at least double-check w/ Mathematica == and FullSimplify) In[10]:= (* your code here * ) 10 10 10 -5 d) Plot the magnetic field at t = 0. (Hints : Use the same SliceVectorPlot3D code to plot it. You may have to play with the VectorPoints and VectorScale options to make it look good. In[11]:= (* your code here *) Extra credit: overlay the E - field and B-field plot by using the Show command. Full credit if the distinction between these fields and their relative alignment can be seen clearly. In[12]:= (* your code here *) e) Does your calculated B-field agree with your sketch? If not, describe and explain the differences. Note carefully the position of the B-field peaks relative to the circular swirls in the E-field. ( Your written answers here ) ctivate Windows