• Homework 1: (Due Friday, October 5)   

  
    For this homework, do not use any vector calculus theorems to simplify the algebra, complete
    the differentiation and integration operations as instructed in the problems.
  
    A tetrahedral volume V is defined by its 4 corners O, A, B and C with the x-y-z coordinates
  
    O : 0, 0, 0
    A : 1, 0, 0
    B : 0, 2, 0
    C : 0, 0, 3
  
    The triangular surfaces S1, S2, S3 and S4 are defined by their corners
  
    S1 : OAB 
    S2 : OAC
    S3 : OBC
    S4 : ABC
  
    with their directions pointing outside V.
  
    1) Find the unit vectors in the directions of S1, S2, S3 and S4.
  
    2) Consider the function v(x,y,z) = xy2z3.
  
       (a) Find its gradient E = ∇ v
  
       (b) Find its laplacian ρ = ∇2 v
  
       (c) Find the curl of E:   C = ∇ x E
  
       (d) Integrate ρ within the volume V:     Q = ∫V ρ d3x
  
       (e) Integrate E along the edge of surface S4:     I = ∮ABCA E ⋅ dl 
  
       (f) Integrate E on the surfaces Si to find its flux through these surfaces: Φi = ∫Si E ⋅ dS
  
       (g) What is the total flux  Φ1 +  Φ2 +  Φ3 +  Φ4 ?
  
  
    3) Consider the vector field A = i xy2z3 + j xyz + k x2yz3
       where i, j and k are the unit vectors in the x, y and z directions respectively.
  
        (a) Find the curl of A :     B = ∇ x A
  
        (b) Find the curl of B :     j = ∇ x B
  
        (c) Find the flux of j through surface S4 : I4 = ∫S4 j ⋅ dS
  
        (d) Integrate B along the edge of surface S4:     i4 = ∮ABCA B ⋅ dl
  
  

• Homework 2: (Due Friday, October 19)

    We are interested in finding the potential within the part of space V given by
  
                          x>0, y>0 and all vales of z.
  
    There is no charge in this region, but the potential at the boundaries is given by
  
          On the surface at y=0: V = Vo  for 0<x<a and 0<z<a
          On all other parts of the boundary: V=0.
  
    In other words, just a square area of dimensions a x a on the x-z plane is at 
    potential V = Vo and all other parts of the boundary are at zero potential.
  
    Find the corresponding Green's function and write down an integral expression for the
    potential at a general point inside V.
  
  

• Homework 3: (Due Friday, December 6)
   
  
   
  

  
    A grounded spherical shell of radius R is centered at the origin.
    A ring charge (with total charge Q) and radius r is centered on the z-axis
    a distance a from the origin.  The plane of the ring is perpendicular to the
    z-axis. Find the potential on the z-axis for -R < z < R. 
  
    Verify your solution for z = ±R.
  
  

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