Fall 2018 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. 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
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.
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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