JEE Main & Advanced Physics Electrostatics & Capacitance Capacitance

(1) Capacitance of a conductor : Charge given to a conductor increases it's potential i.e.,  \[Q\propto V\] \[\Rightarrow \] \[Q=CV\]

Where C is a proportionality constant, called capacity or capacitance of conductor. Hence capacitance is the ability of conductor to hold the charge.

(2) It's S.I. unit is \[\frac{Coulomb}{Volt}=Farad (F)\]

Smaller S.I. units are \[mF,\,\mu F,\,\,nF\] and pF (\[1mF={{10}^{-3}}F\], \[1\mu F={{10}^{-6}}F\], \[1nF={{10}^{-9}}F\], \[1pF=1\mu \mu F={{10}^{-12}}F\])

(3) It's C.G.S. unit is Stat Farad \[1F=9\times {{10}^{11}}\,Stat\,\,Farad\].

(4) It's dimension : \[[C]=[{{M}^{-1}}{{L}^{-2}}{{T}^{4}}{{A}^{2}}]\].

(5) Capacity of a body is independent of charge given to the body or it's potential raised and depends on shape and size only.

(6) Capacity of an isolated spherical conductor : When charge Q is given to a spherical conductor of radius R, then potential at the surface of sphere is \[V=\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{Q}{R}\] Þ \[\frac{Q}{V}=4\pi {{\varepsilon }_{0}}R\]

\[C=4\pi {{\varepsilon }_{0}}R=\frac{1}{9\times {{10}^{9}}}.R\]

If earth is assumed to be a conducting sphere having radius \[R=6400\,\,km.\] It's theoretical capacitance \[C=711\,\mu F\]. But for all practical purpose capacitance of earth is taken infinity and its potential \[V=0\]. (7) Energy of a charged conductor : Electrostatic potential energy of a conductor carrying charge Q, capacitance C and potential V is given by  \[U=\frac{1}{2}QV=\frac{1}{2}C{{V}^{2}}=\frac{{{Q}^{2}}}{2C}\]