Question 25.41: A coaxial cable used in a transmission line has an inner rad......
A coaxial cable used in a transmission line has an inner radius of 0.10 mm and an outer radius of 0.60 mm. Calculate the capacitance per meter for the cable. Assume that the space between the conductors is filled with polystyrene.
THINK Our system, a coaxial cable, is a cylindrical capacitor filled with polystyrene, a dielectric.
EXPRESS Using Eqs. 25-17 and 25-27, the capacitance of a cylindrical capacitor can be written as
C=4 \pi \varepsilon_0 \frac{a b}{b-a} (spherical capacitor). (25-17)
C=\kappa \varepsilon_0 \mathscr{L} =\kappa C_{\text {air }} (25-27)
C=\kappa C_0=\frac{2 \pi \kappa \varepsilon_0 L}{\ln (b / a)},
where C_0 is the capacitance without the dielectric, \kappa is the dielectric constant, L is the length, a is the inner radius, and b is the outer radius.
ANALYZE With \kappa = 2.6 for polystyrene, the capacitance per unit length of the cable is
\frac{C}{L}=\frac{2 \pi \kappa \varepsilon_0}{\ln (b / a)}=\frac{2 \pi(2.6)\left(8.85 \times 10^{-12}\, F / m \right)}{\ln [(0.60\, mm ) /(0.10 \,mm )]}=8.1 \times 10^{-11} \,F / m =81 \,pF / m .
LEARN When the space between the plates of a capacitor is completely filled with a dielectric material, the capacitor increases by a factor \kappa, the dielectric constant characteristic of the material.