Fall 2021 Homework Assignments
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 have 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
The three dimesional region of space for which x>0 and y>0 is bounded by the surfaces at x=0 and y=0 and infinity. The surface at y=0 is maintained at a constant potential Vo while the other boundaries are at zero potential. Using the Green's function method, find the potential inside this region.
The cross section of an infinitely long ctlindrical structure is shown in the figure. The potential is specified as shown at radii a and b. Find the expression for the potential at radii a < r < b. Putting in numerical values for a, b, and Vo, obtain a 3-D plot of the potential as a function of position. Your result could resemble this:
A thin disk of radius R carries a uniformly distributed total charge Q. The disk is placed on the x-y plane, centered at the origin. Find the cartesian multipole expansion of the potential due to this charge distribution.
A dielectric sphere of radius R is placed at the origin. An electric dipole of magnitude Po is placed inside the sphere, on the z-axis, at z=a, , pointing in the z-direction. Find the electric potential at all points in space. Find its asymptotic form at large distances from the origin.
A wire in the shape of a square, carrying a current I is placed on the x-y plane as shown in the figure. Find the Ax(r,θ,φ), Ay(r,θ,φ) and Az(r,θ,φ) components of the vector potential at a general point (r,θ,φ) with r>a, as an expansion in spherical harmonics. Keep sufficient number of terms in the expansion to see the effects of the square shape. Hint: Use the form of the spherical harmonics in rectangular coordinates for the integral over source coordinates.
Each homework is to be placed as a single PDF document (which may contain photos of your work) and submitted through moodle.
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Check the Stars system for your homework grade. In some cases, you may be able to re-submit your homework to the assistant with corrections and improve your grade.