Capacitance of a Co-axial cable

A Co-axial cable is a Transmission line, in which two conductors are placed co-axially and are separated by some dielectric material with dielectric constant (or) permittivity  (\epsilon ).

a conductor is in the form of a cylinder with some radius, let the radius of inner conductor is ‘a’ meters and that of outer conductor be ‘b’ meters.

Now connect this co-axial conductor to a supply of ‘V’ volts , after applying ‘V’ assume positive charges are distributed on M_{2} and negative charges on M_{1} .

Now, a field is induced \overrightarrow{E} between M_{2}  and M_{1} because of flux lines, to find out \overrightarrow{E} at any point P  between these two conductors

location of P is out of the conductor M_{2} an inside the conductor M_{1}.

\therefore \overrightarrow{E}_{at P} = \overrightarrow{E}_{\ due \ to \ inner \ conductor \ M_{1}} .

assume a cylindrical co-ordinate system \rho ,\ \phi , \ z  and axis of cable coincides with z-axis this is similar to a line charge distribution \rho_{L} placed along the z-axis.

\rho _{L}=\frac{Q}{L} .

\therefore \overrightarrow{E}_{at P} = \frac{\rho_ {L}}{2\pi \epsilon _{o}\rho }\overrightarrow{a}_{\rho } .

\therefore V = -\int_{1}^{2}\overrightarrow{E}.\overrightarrow{dl} .

V = -\int_{1}^{2} \frac{\rho_ {L}}{2\pi \epsilon _{o}\rho }\overrightarrow{a}_{\rho }.(d\rho \overrightarrow{a}_{\rho }+d\phi \overrightarrow{a}_{\phi }+dz \overrightarrow{a}_{z }) .

V = -\int_{b}^{a} \frac{\rho_ {L}}{2\pi \epsilon _{o}\rho }\overrightarrow{a}_{\rho }.(d\rho \overrightarrow{a}_{\rho }) .

V = - \frac{\rho_ {L}}{2\pi \epsilon _{o} }(\ln a-\ln b) .

V = \frac{\rho_ {L}}{2\pi \epsilon _{o} }(\ln b-\ln a) .

V = \frac{\rho_ {L}}{2\pi \epsilon _{o} }\ln (\frac{b}{a}) .

V = \frac{Q}{2\pi \epsilon _{o}L }\ln (\frac{b}{a}) \ \because \ \rho _{L} = \frac{Q}{L} .

\therefore C_{co-axial} =\frac{Q}{V} = \frac{2\pi \epsilon_{o}L}{\ln (\frac{b}{a})} .

L- length of the conductors.

b-radius of the outer conductor.

a- radius of the inner conductor.

\epsilon – permittivity of the medium.

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Author: vikramarka

Completed M.Tech in Digital Electronics and Communication Systems and currently working as a faculty.