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 are having 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 dipole p = p n is placed on the z-axis at the coordinate (0,0,a). n is the unit vector (i + j)/√2. A point charge Q is also placed at the origin. (a) Find the electric field due to the point charge at the position of the dipole. (b) Find the electric field due to the dipole at the position of the point charge. (c) Find the force acting on the point charge. (d) Find the force acting on the dipole. (e) Find the torque acting on the dipole. (f) Find the total force and the total torque on the system.
A line charge with charge density λ per unit length is placed on the z-axis extending
from -a/2 to a/2.
(a) Find the electric field (magnitude and direction) at a point on the z-axis for z>a.
Find the asymptotic form of this quantity as z→∞.
(b) Find the electric field (magnitude and direction) at a point on the x-axis.
Find the asymptotic form of this quantity as x→∞.
(c) Find the asymptotic form for x≪a.
A thin conducting shell of radius R is placed at the origin. Two point charges q (at x=a) and
Q (at x=b) are placed on the x-axis with a < R < b. The shell carries zero total charge.
(a) What is the electric field at the origin?
(b) What is the potential of the sphere?
(c) Find the forces on both charges and the conductor.
Find the potential in a cylindrical region bounded at radii a and b, with a<b. The potential at radius ρ=a is given by Va(φ)=Vo cos(φ) and at ρ=b is given by Vb(φ)=Vo sin(2φ). This is the potential distribution I find for b=2a:
Find the potential in a spherical region bounded at radii a and b, with a<b. The potential at radius r=a is given as Va=0 and at r=b is given by Vb(θ)=Vo cos(2θ). Hint: Insert x=cos(θ) in the expression for P2(x), and expand it to determine cos(2θ) in terms of P0(x) and P2(x). This is the potential distribution I find for b=2a:
1. By directly integrating the expression for the potential V(r) = 1/(4πεo) ∫d3r' ρ(r') /|r-r'|,
obtain the potential, on the z-axis, due to a charged disk with radius R and carrying a total
charge Q uniformly distributed on its surface. (The disk is centered at the origin and lies
on the x-y plane.) Expand this expression for large z and verify that you get the same coefficients
as the expansion in terms of Legendre polynomials that we got in class.
2. Find all components of the quadrupole moment Qij for the charge densities given in the figure.
(Note that since many of the components will be related due to symmetry, you will not have to
calculate all explicitly.)
What is the behavior of the E-field in the +x, +y, and +z axes due to the quadrupole moment?
1. A polarized sphere of radius R is known to have a polarization density in form p⃗ = po z/R 𝕫 where po is a constant and 𝕫 is the unit vector in the z-direction. (a) Find the bound volume and surface charge densities ρb and σb. (b) Find the potential distribution corresponding to the bound surface charge density only. 2. A very long cylindrical pipe with inner radius a and outer radius b has dielectric permeability ε. The cylinder is placed coaxially with the z-axis. An electric field which is uniform at large distances from the z-axis (with magnitude Eo, in the x-direction) is also present. (a) Find the potential at all points in space. (b) What is the electric field on the z-axis? Discuss the large and small ε limits.
1. A square loop is placed on the x-y axis as shown in the figure. A current I flows along the loop.
A uniform magnetic field in the x-direction, with magnitude Bo is also present.
(a) Find the magnitude and direction of the forces acting on the four parts of the loop.
(b) Find the torque acting on the loop.
2. (a) Find the magnetic field, on the z-axis, generated by the current in the loop in the figure.
Hint: Use the approach we used in class to find the magnetic field due to an infinite
straight wire, but limit your integral to the finite length of one side of the square.
You will then add the fields due to the 4 sides - symmetry of the problem is your friend.
(b) Find the asymptotic form of this field for very large z. See if this form is consistent
with the electric field of an electric dipole.
