Electrostatics. Selected problems with solutions.

Electrostatics: Selected Problems with Solutions



1 Mathematical Background: The Cylindrical and Spheri-
cal Co-ordinate Systems
In electrostatics, spherical or cylindrical symmetry is present in many problems. Using spher-
ical and cylindrical coordinates rather than the standard Cartesian coordinates can simplify the
computations significantly.

1.1 Cylindrical co-ordinate system
Points are specified on the surface of a cylinder in this kind of coordinate system. Here we use
the co-ordinates: ρ, φ and z. See the figure.




Figure 1: The Circular-cylindrical co-ordinate system

ρ is called the radial co-ordinate, φ is called the azimuthal co-ordinate and z is the usual z
co-ordinate. The projection of point P on the xy plane has the co ordinates (ρ, φ ).
In circular cylindrical co-ordinates, a point P is specified by the curvilinear co-ordinates, ρ, φ
and z. The unit vectors along ρ, φ , z are ρ̂, φ̂ , and ẑ respectively.




1

, They are related to the usual Cartesian co-ordinates by the relations:

x = ρ cosφ
y = ρ sinφ
z=z

Also we notice that
p
ρ= x2 + y2
y
φ = tan−1 ( )
x
z=z

Elemental length in cylindrical co-ordinates is given by the expression

ds2 = dρ 2 + ρ 2 dφ 2 + dz2

A point P in a cylindrical co-ordinate system is the intersection of three surfaces (the plane
surfaces (φ = const and z = const), and the cylindrical surface ρ = const). The differential
volume element dV at P has side lengths ρdφ dz perpendicular to the radial direction, dρdz
perpendicular to the azimuthal direction, and ρdφ dρ perpendicular to the z direction as shown
in the figure below.




Figure 2: Area and Volume elements

The differential normal area is


dS1 = ρdφ dz ρ̂
dS2 = dρdz φ̂
dS3 = ρdφ dρ ẑ

2

, The elemental displacement is in cylindrical co-ordinates is

dl = dρ ρ̂ + ρdφ φ̂ + dz ẑ

The infinitesimal volume element is

dV = ρdρdφ dz

To convert a vector in Cartesian co-ordinates to that in cylindrical co-ordinates we use:
    
Aρ cosφ sinφ 0 Ax
    
    
Aφ  = −sinφ cosφ 0 Ay 
    
    
Az 0 0 1 Az

The gradient of a function ψ in cylindrical co-ordinates is

∂ψ 1 ∂ψ ∂ψ
∇ψ = ρ̂ + φ̂ + ẑ
∂ρ ρ ∂φ ∂z
The expression for divergence in cylindrical co-ordinates is

1 ∂ 1 ∂ Fφ ∂ Fz
∇.F = (ρFρ ) + +
ρ ∂ρ ρ ∂φ ∂z
The expression for curl in cylindrical co-ordinates is i.e.,
     
1 ∂ Fz ∂ Fφ ∂ Fρ ∂ Fz 1 1 ∂ Fρ
∇×F = − ρ̂ + − φ̂ + Fφ − ẑ
ρ ∂φ ∂z ∂z ∂ρ ρ ρ ∂φ

This can be written in compact form as:

ρ̂ ρ φ̂ ẑ
1 ∂ ∂ ∂
∇×F = ∂ρ ∂φ ∂z
ρ
Fρ ρ Fφ Fz

The Laplacian in cylindrical co-ordinates is:

∂ 2ψ 1 ∂ 2ψ ∂ 2ψ 1 ∂ ψ
∇2 ψ = + + 2 +
∂ ρ2 ρ2 ∂ φ 2 ∂z ρ ∂ρ
The position vector of a point in cylindrical coordinate can be written as:

r = ρ ρ̂ + z ẑ (1)



3

, 1.2 Spherical co-ordinate system
Here the points are defined on the surface of a sphere. We specify any point P by three co-
ordinates u = r, v = θ , w = φ , where r = OP is the distance of the point P from the origin, θ
is the angle between OP and the z-axis and φ is the angle between the planes xz and OPZ.




Figure 3: Spherical Polar Co ordinate System

The spherical co-ordinates can be transformed to Cartesian co ordinates by the relations

x = r sinθ cosφ = ρ cosφ
y = r sinθ sinφ = ρ sinφ
z = r cosθ

where ρ = r sinθ is the projection of the position vector OP on xy plane. The unit vectors
along r, θ , φ are r̂, θ̂ and φ̂ respectively.
Also, we have
p
r = x2 + y2 + z2
p !
x 2 + y2
θ = tan−1
x
y
φ = tan−1
x
Elemental length in spherical co-ordinates is given by the expression

dS2 = dr2 + r2 dθ 2 + r2 sin2 θ dφ 2

4