Uniqueness theorem, boundary conditions, boundary value problems

Uniqueness theorem

Problem:

Prove, carefully explaining your reasoning, that the solution of E = ρ/ε0,×E = 0, for E is unique.  You may assume that the sources of E are bounded in space and that therefore the field vanishes at sufficiently large distances from the sources.

Solution:


Boundary value problems, azimuthal symmetry

Problem:

A sphere of radius R carries a surface charge density σ = σ0cosθ.
(a)  Derive the exact potential everywhere (both inside and outside the sphere) assuming that it vanishes as r goes to infinity.
(b)  Calculate the dipole moment of the charge distribution and deduce the approximate form of the potential at points far from the sphere (r >> R).
(c)  Compare (a) and (b).  What can you conclude about the higher multipoles?

Solution:

Problem:

A hollow ball of radius R has a surface charge distribution which produces a potential on the surface of V(R,θ) = k(cos2θ + 1), where k is a constant and θ is the usual polar angle relative to the z-axis.
(a)  Find the potential at all points in space.
(b)  Show that the charge distribution on the ball is σ(R,θ) = (kε0/R)(5cos2θ - ⅓).

Solution:

Problem:

Consider a hollow sphere of radius R with a surface potential Φ(R,θ) = k sin2θ, where k is a constant and θ is the usual polar angle relative to the z-axis.
Find the potential everywhere inside the sphere.
Hint:
P0(x) = 1
P1(x) = x
P2(x) = (3x2 – 1)/2
P3(x) = (5x3 – 3x)/2

Solution:

Problem:

Two thin wires, each of length L, lie on the z-axis.  Wire 1 has a positive charge per unit length +λ and extends from (0, 0, L) to (0,0,0).  Wire 2 has a negative charge per unit length -λ and extends from (0,0,0) to (0,0,-L).
image
(a)  Find the potential Φ on the z-axis at z > L.
(b)  The potential Φ(r,θ) at a arbitrary points in space with r > L can be expanded in terms of Legendre polynomials,  Φ(r,θ) = ∑n=0[Anrn + Bn/rn+1]Pn(cosθ). 
Find the expansion coefficients.
Note: ln(1 + x) = ∑n=1(-1)n+1xn/n

Solution:

Problem:

A charge array consists of two charges, each of magnitude +q, located on the z-axis at (0, 0, ±a).
(a)  Find the potential V(0,0,z) at an arbitrary point z > a; then expand V(0, 0,z) in a power series in z.
(b)  Using this series as a “boundary condition”, find the potential V(r,θ,φ) at an arbitrary location (r,θ,φ) with r > a.  An infinite series is acceptable.
(c)  Characterize the first three terms in 1/rn.

Solution: