Question

a signal is carried by a coaxial cable. The signal is carried by an inner wire with radius R1 while an outer conductor of radius R2 is grounded. An insulating materiel is between the two.

A. Find an expression for the capacitance per meter of the coaxial cable, assuming air is between the two cylinders.

B. Find the Capacitance per meter of a cable with R1= 0.70mm and R2=2.1mm. (answer units are pF/m)

Answer 1

A )

using

= E.ds

E.ds = q/_o

E = q / 2 _o r L

V = - E dr

= - ( q / 2 _o L ) ( 1/r ) dr

V = ( q / 2 _o L ) ln ( R₁/R₂ )

C = Q / V

= Q / ( q / 2 _o L ) ln ( R₁/R₂ )

= 2 _o L / ln ( R₂/R₁ )

C/L = 2 _o / ln ( R₂/R₁ )

C = 2 _o / ln ( R₂/R₁ ) ( L = 1 m given )

so the expression for the capacitance per meter of the coaxial cable, assuming air is

between the two cylinders C = 2 _o / ln ( R₂/R₁ )

B )

given

R₁ = 0.70 mm

= 0.7 x 10^-3 m

and

R₂ = 2.1 mm

= 2.1 x 10^-3 m

C/L = 2 _o / ln ( R₂/R₁ )

= 2 x 3.14 x 8.85 x 10^-12 / ln ( 2.1 x 10^-3 / 0.7 x 10^-3 )

= 2 x 3.14 x 8.85 x 10^-12 / ln ( 2.1 / 0.7 )

C/L = 5.0589 x 10^-11 F/m

= 50.589 x 10^-12 F/m

so the Capacitance per meter of a cable is C/L = 50.589 pF/m