A cylindrical capacitor has two co-axial cylinders of length 15 cm and radii 1.5 cm and 1.4 cm. The outer cylinder is earthed and the inner cylinder is given a charge of 3.5 µC. Determine the capacitance of the system and the potential of the inner cylinder. Neglect end effects (i.e., bending of field lines at the ends).

#### Solution

Length of a co-axial cylinder, l = 15 cm = 0.15 m

Radius of outer cylinder, r_{1} = 1.5 cm = 0.015 m

Radius of inner cylinder, r_{2} = 1.4 cm = 0.014 m

Charge on the inner cylinder, q = 3.5 µC = 3.5 × 10^{−6} C

Capacitance of a co-axial cylinder of radii r_{1} and r_{2} is given by relation

C = `(2piin_0"l")/log_"e"("r"_1/"r"_2)`

Where,

`in_0` = Permittivity of free space = `8.85 xx 10^-12 "N"^-1 "m"^-2 "C"^2`

∴ C = `(2pi xx 8.85 xx 10^-12 xx 0.15)/(2.3026 log_10(0.15/0.14))`

= `(2pi xx 8.85 xx 10^-12 xx 0.15)/(2.3026 xx 0.0299) = 1.2 xx 10^-10 "F"`

Potential difference of the inner cylinder is given by,

`"V" = "q"/"C"`

= `(3.5 xx 10^-6)/(1.2 xx 10^-10) = 2.92 xx 10^4 "V"`