Cylindrical Capacitor

A capacitor made up of two concentric cylindrical shell or wires separated to each other by a dielectric in between them as shown in the figure below is known as the Cylindrical Capacitor.

Capacitance of Cylindrical Capacitor:

Let, The radius of the inner cylinder be R1 and that of the outer cylinder be  R2. And let the relative permittivity of the medium in between the two cylinders be \epsilon_r .

Let the charge per meter length of the cable on the outer surface of the inner cylinder be +Q coulomb and on the inner surface of the outer cylinder be -Q coulombs. for all practical purpose, the charge of +Q coulombs/m on the surface of the core can be assumed to be located at it’s axis. The outer cylinder is earthed.

Cylindrical Capacitor
Cylindrical Capacitor

 

Now,

Surface area of the coaxial cylinder of radius x meters and length of one meter is 2 \pi x m2.

Thus, The electrical field intensity at a point x meters away from the center of the inner cylinder is:

E_x = \dfrac{Q}{2 \pi \epsilon_0 \epsilon_r x}

or,
E_x = \dfrac{Q}{2 \pi \epsilon_0 \epsilon_r}\dfrac{1}{ x} V/m

Thus, The potential difference developed between the two plates or cylinders in the parallel plate capacitor (V) is:

V = \displaystyle{\int_{R_1}^{R_2}-E dx}

or,
V = \displaystyle{\int_{R_1}^{R_2}} \dfrac{-Q dx}{2 \pi \epsilon_0 \epsilon_r x}

or,
V = \dfrac{-Q}{2 \pi \epsilon_0 \epsilon_r } \displaystyle{\int^{R_2}_{R_1} \dfrac{dx}{x}}

or,
V = \dfrac{-Q}{2 \pi \epsilon_0 \epsilon_r }\log_e R_1 - \log_e R_2

or,
V = \dfrac{-Q}{2 \pi \epsilon_0 \epsilon_r } \log_e \left( \dfrac{R_1}{R_2}\right)

or,
\dfrac{Q}{V} = \dfrac{2 \pi \epsilon_0 \epsilon_r }{\log_e \left( \dfrac{R_2}{R_1}\right)}

Or, the capacitance C in the case is:
C = \dfrac{2 \pi \epsilon_0 \epsilon_r }{\log_e \left( \dfrac{R_2}{R_1}\right)} F/m

Thus, The capacitance of a cylindrical capacitor with L meter length or height and R_1 , R_2 the radius of inner and outer cylinder respectively is:
C = \dfrac{2 \pi \epsilon_0 \epsilon_r L }{\log_e \left( \dfrac{R_2}{R_1}\right)} F

Cylindrical Capacitor Calculator:

Here is a simple Cylindrical Capacitance calculator. Note; You will need to enable javascript for this calculator to work which is enabled in most browsers by default.

Relative Permittivity of the medium( e):
Length or height of the Cylindrical Capacitor(L in meters):

Radius of the inner cylinder from axis(R1 In meters)

Radius of the outer cylinder from axis(R2 In meters)

Resultant Capacitance in farad: 
Resultant Capacitance in microfarad: 
Resultant Capacitance in picofarad: