Concentric Cylinders

The conductor and sheath of coaxial cables, such as telegraph cable and single-conductor lead-sheath power cable, are concentric cylinders. The capacitance between the inner cylinder, the conductor, and outer cylinder, the sheath or shield, has an important effect on the characteristics of such cables. Consider the two concentric cylinders in Fig. 2-15(a).

Let

r₁ = the radius of the inner cylinder in meters
r₂ = the radius of the outer cylinder in meters
l = length of the cylinders in meters
q = charge on the inner cylinder in coulombs

[2-34]

In the elemental shell of radius x and thickness dx

[2-35]

The charge q on the inner cylinder is stationary. Therefore, the electric field intensity must be normal to the surface of the conductor because any tangential component or axial component would cause a flow of current in a tangential or an axial direction.

Figure 2-15. (a) Concentric cylinders; (b) single-conductor cable.

The angle between E and dx is zero, therefore

The total voltage between the inner and outer cylinders is

[2-36]

but C = q/V, hence

[2-37]