Capacitance of a Co-axial cable

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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.

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

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

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

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