Fall 2022 Homework Assignments
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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 (with charge λ per unit length) is placed on the z-axis and extends from z=-a/2 to z=a/2.
(a) Find the electric field at a general point (x,z) on the x-z plane.
(b) Find the asymptotic form of this result for
(i) x≫a on the x-axis
(ii) x≪a on the x-axis
(iii) z≫a on the z-axis
A conductive sphere with radius R is placed at the origin. The sphere has zero net charge on it. Two point charges with magnitudes Q and q are placed on the +z axis at points a and b with a>b>R. Find the energy stored in the system.
A rectangular box encloses the space for 0<x<a and 0<y<b. (There is no variation in the z-direction.) All surfaces of the box are kept at zero potential. A flat surface charge density parallel to the y-z plane is placed at x=a/2 inside the box. The charge density depends on the y-coordinate and has the form σ(y) = σo sin(2πy/b) where σo is a constant. Find the potential at a general point inside the box.
(a) Two concentric spheres have the following potentials on them: The inner sphere with radius a
has potential V(θ) and the outer sphere with radius b has -V(θ) as a function
of the polar angle θ. The potential has the expansion
V(θ) = Σℓ Cℓ Pℓ(cos θ) .
The coefficients Cℓ for expansion in terms of the Legendre polynomials Pℓ(cos θ) are "given".
Find the potential V(r,θ) in the region of space for a < r < b.
(b) A line charge (with charge λ per unit length) is placed on the z-axis and extends from z=-a/2 to z=a/2.
(i) Find the potential V(z) on the z axis for z > a/2.
(ii) Expand this potential in powers of z to fourth order in z.
(iii) From the expression in (ii), obtain the potential at (r,θ) to fourth power in r.
A spherical dielectric with radius R is centered at the origin and it is known that it
has a polarization in the z-direction, proportional to the z-coordinate: P = α z k
where α is a constant and k is the unit vector in the z-direction.
(a) Express z and k in terms of the spherical coordinate variables r, θ and φ and the
unit vectors associated with them.
(b) Find the divergence of P in rectangular and spherical coordinate systems.
(c) What are the volume and surface charge densities associated with this polarization?
(d) Find the potential inside and outside the sphere.
[Hint: You will have to solve the problem separately for the volume and the surface charge
densities and then combine the two solutions.]
Each homework is to be placed as a single PDF document (which may contain photos of your work) and submitted through moodle.
Please name the file in the format:
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.