P.21 |
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P.23 Coaxial Capacitor |
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Statement
A shielded power cable has a polyethylene insulation with specified relative permittivity and a dielectric strength of 18.1 MV/m. What is the upper limit of voltage on the inner conductor with respect to the shield when the inner conductor has a radius r1 and the inner side of the concentric shield is at radius r2?
System Parameters
Dielectric strength:
Permittivity of free space:
Solution
To solve this problem we will need to find the capacitance of a cylindrical system. With fringing neglected, Gauss' law requires that D be inversely proportional to r between the conductors (see Problem 6.6). Assuming some positive surface charge density on the inner conductor,
to meet the boundary condition at the surface of the conductor
The voltage difference between the conductors is, then,
Choose Evaluate Symbolically from the Symbolic menu (if you are using the Mathcad Engine, you won't have this option) to give
from which the capacitance per unit length is
or
in this case
To solve for the maximum voltage, the capacitance and the charge are required. Since the maximum electric field will be equal to the dielectric strength, and the maximum field strength occurs on the inner conductor (smallest radius = largest electric field), solve for the charge density on the inner conductor using this equality:
The total charge per meter is
and therefore voltage is given by
