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Question
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, r1 = 1.5 cm = 0.015 m
Radius of inner cylinder, r2 = 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 r1 and r2 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"`
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