• Homework 1: (Due Monday, Sep. 28)   

  
    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.
  
    This first homework is intended to remind you of the vector calculus operations
    we will use throughout this course. If you are having difficulty carrying out the
    operations, I reccommend a fast review of the relevant topics.
  
  
    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 : 2, 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) = x3y2z2.
  
       (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 x2yz + j xy2z2 + k x3yz2
       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 Monday, Oct 5)

  (a) In class, we found the potential on the axis of a ring charge placed on the x-y plane. Using this result,
      find the potential on the z-axis due to two ring charges, both placed parallel to the x-y plane,
      with radius R, and centered on the z-axis. One of the rings is centered at z=a and carries charge +Q.
      The second ring is centered at z=-a and carries charge -Q.
      
  (b) Sketch your result as a function of z, for all z.
  
  (c) Find the asymptotic behavior of this potential for large values of z. (i.e. for z → ±∞)
  

• Homework 3: (Due Monday, October 19)
  A dipole \( \vec p_1 = p_1 \; \hat x\) is placed at the origin. A second dipole \( \vec p_2 \) is placed on the \(x\)-axis at $x=R$.
  \( \vec p_2 \) lies on the \(x\)-\(y\) plane and makes an angle \( \alpha \) with the \( x\)-axis.
  Find the torque and the force on \( \vec p_2 \) as a function of \( \alpha \).
  Comment on the special cases \( \alpha = 0,\) π/2, and π.

• Homework 4: (Due Monday, October 26)
  A dipole \( \vec p_1 = p_1 \hat z\) is placed on the \(z\)-axis at \(z=a\). A spherical conductor with zero potential and
  radius \(R\) with \(R\)<\(a\) is centered at the origin.
  Find the image of the dipole inside the sphere. (Note that the image is not simply another dipole.)
  [Hint: Treat the dipole as two opposite point charges a small distance apart,
  find their images, and then analyze the point-dipole limit.]

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