Fall 2019 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
A line charge is placed inside a spherical conductor (radius R) which is centered at the origin. The line charge (with uniform charge λ per unit length) extends on the z-axis from the origin to z=R/2. (a) Find the location and the charge density (charge per unit length) distribution of the image charge. (b) Find the force between the line charge and the conductor.
A two dimensional geometry (no variation in z-direction) is defined in the region 0<x<a and -∞<y<∞. The region is charge-free but has the following boundary conditions at x=0 and x=a boundaries: { Vo for 0 < y < b/2 V(x=0,y) = V(x=0,y+b) with V(x=0,y) = { { -Vo for b/2 < y < b and { -Vo for 0 < y < b/2 V(x=a,y) = V(x=a,y+b) with V(x=a,y) = { { Vo for b/2 < y < b . That is, the potential at the boundaries are periodic with length b, and are shifted with respect to one another by half a wavelength. (a) Find the potential V(x,y) within the geometry. Use the symmetries in the problem to simplify your algebra. (b) Obtain plots of V(x,y) as a function of y/b for values of x=0, a/4, a/2, and 3a/4.
An electric dipole with dipole monent p is pointing in the x-direction and is placed on the z-axis at z=a. Expand the potential in terms of the variables r, θ and φ to l=4 order in terms of the spherical harmonics Ylm.
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.