- Homework 1:
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
- Homework 2: (Due October 6th)
An infinite line charge (with charge λ per unit length) is placed on the z-axis.
A second infinite line charge (with charge -λ per unit length) is placed parallel to it at x=a and y=0.
(a) Find the electric potential at a general point (x,y,z).
(b) Go to a "line-dipole" limit where a→0 and λ→∞ such that the potential stays finite.
(c) Differentiate this potential so that you find the electric field at the point (x,y,z).
(d) Repeat part (c) in cylindrical coordinates to find the field at (ρ,z,φ).
- Homework 3: (Due October 13th)
A point dipole of magnitude p and pointing in the +z direction is placed on the z-axis at z=a.
A grounded sphere (zero potential) with radius R (with R<a) is placed at the origin.
Represent the dipole as two opposite charges with a small distance between them. Find the images
corresponding to these charges and evaluate them at the point dipole limit. Check that the potential
goes to zero at the surface of the sphere, at z=R and z=-R.
Hint: Notice that the image of the dipole is not simply another dipole.
- Homework 4: (Due October 20th)
A right-triangular area shown in the figure with perpendicular sides of equal lengths a has
potential zero on its diagonal side. The potential along the x-axis is given as a contant Vo
and as sin(2πy/a) along the y-axis. Find the potential inside the triangular region.
Hint: Use the symmetry of the triangle to extend the region to one you can more easily solve.
- Homework 5: (Due December 1st)
A spherical conductor centered at the origin and radius D is kept at zero potential.
It is covered with a material with permittivity ε up to radius R. A point charge Q
is placed a distance a from the origin on the z-axis. Find the potential at all points in space
for radius r > D.
- Homework 6: (Due December 8th)
A wire in the shape of a square loop carries a current I. The sides of the square are of length a.
The wire is centered at the origin and is plced on the x-y plane. Find the magnetic field on the
z-axis. Find its limiting behavior at z=0 and z→∞.
- Homework 7: (Due December 15th)
Find the vector potential due to the loop of wire described in Homework 6, at a point on the
x-z plane, at large distances from the loop.
- Homework 8: (Due December 22nd)
Two magnetic dipoles, each of strength μ are placed on the x-y plane.
The first dipole is placed at the origin and points in the +z direction.
The second dipole is placed on the x-axis at the coordinate (a,0,0) and
points in the +x direction.
Find the forces and torques on both dipoles and show that all the torques
and the forces internal to the system add up to zero.
Please send your homework to the assistant as a single PDF
file. You can use the printer/copier on the Physics Department floor to
scan your handwritten papers into PDF or put photos into a file using your favorite editor.