- Cylindrical Capacitor Definition: A cylindrical capacitor is defined as a type of capacitor with a central conductor, an insulating layer, and an outer grounded metallic cover.
- Power Cable Example: Electrical power cables act as cylindrical capacitors with their conductor, insulation, and outer grounded cover.
- Charge Distribution: Charge per meter length (Q coulombs) in the cable is used in capacitance calculations.
- Capacitance Formula: The capacitance per unit length is calculated using the radii of the conductor and the cable.
- The capacitance per unit length is calculated using the radii of the conductor and the cable.: Understanding cylindrical capacitors helps in analyzing the capacitance in power cables.
Capacitance of a Cylindrical Power Cable
A cylindrical capacitor can be modelled by a long coaxial Electrical Power Cable. The inner conductor and outer metallic screen or sheath are the two electrodes, separated by insulation. The derivation assumes concentric cylinders, a homogeneous dielectric and negligible end effects.
A voltage difference, rather than current flow, establishes equal and opposite charge on the two conductors. Let the charge per metre on the inner conductor be Q coulombs. Its outer radius is r1, and the inner radius of the outer conducting screen is r2.
To derive the capacitance per unit length of this cylindrical capacitor, choose a coaxial Gaussian cylinder of radius x between the two conductors.
For a one-metre length, the curved surface area of that Gaussian cylinder is:
Gauss’s law relates the enclosed charge to electric flux density on this surface:
Dividing flux density by dielectric permittivity gives the radial electrical field intensity at radius x:
The potential difference follows from integrating the radial field. Its differential relation to voltage is:
Integrate between the inner-conductor radius r1 and the outer-screen inner radius r2:
Here V1 is the potential at r1, while V2 is the potential at r2.
If the outer conductor is grounded, V2 = 0 and the conductor-to-screen voltage is:
Using charge divided by voltage, the cable capacitance per metre is:
Thus C per unit length equals 2π times the dielectric permittivity divided by the natural logarithm of r2/r1.
For a uniform cable of length L with negligible end effects, capacitance scales linearly with L:
