Electrostatic Potential and Capacitance
Combination of Capacitors and Energy Stored in a Capacitor
 When capacitors one connected in series all plates have same charge in magnitude.
 Potential differences \tt V_{1}:V_{2}:V_{3}=\frac{1}{C_{1}}:\frac{1}{C_{2}}:\frac{1}{C_{3}}
 Equivalent capacitance \tt \frac{1}{C}=\frac{1}{C_{1}}+\frac{1}{C_{2}}+\frac{1}{C_{3}}
 When ‘n’ identical capacitors one one connected in series \tt C_{s}=\frac{C}{n}
 Energy of capacitors \tt U_{1}:U_{2}:U_{3}=\frac{1}{C_{1}}:\frac{1}{C_{2}}:\frac{1}{C_{3}}
 Total energy of the combination U = U_{1} + U_{2} + U_{3}
 In parallel combination of capacitors potential is same
 Total charge is Q = Q_{1} + Q_{2} + Q_{3} +     
 The Ratio of charges in parallel combination Q_{1} : Q_{2} : Q_{3} = C_{1} : C_{2} : C_{3}
 Equivalent capacitance in parallel C = C_{1} + C_{2} + C_{3} +     
 “n” identical capacitors in parallel Cρ = nc.
 Energy of capacitors U_{1} : U_{2} : U_{3} = C_{1} : C_{2} : C_{3}
 Total Energy of combination in parallel U = U_{1} + U_{2} + U_{3}
 Two dielectrics K_{1} , K_{2} in parallel between capacitors then \tt C_{eff}=\frac{\varepsilon_{0}A}{d}\left(\frac{K_{1}+K_{2}}{2}\right)
 Two dielectrics K_{1}, K_{2} in series between capacitors the \tt C_{eff}=\frac{\varepsilon_{0}A}{d}\left(\frac{2K_{1}K_{2}}{K_{1}+K_{2}}\right)
 Two capacitors C_{1} C_{2} charged to V_{1} V_{2} potentials are connected in parallel then common parallel \tt V=\frac{Q_{1}+Q_{2}}{C_{1}+C_{2}}=\frac{C_{1}V_{1}+C_{2}V_{2}}{C_{1}+C_{2}}
 Loss of Energy = \tt \frac{1}{2}\ \frac{C_{1}C_{2}}{C_{1}+C_{2}}\ \left(V_{1}V_{2}\right)^2
 Capacity of spherical capacitor \tt C=4\pi \varepsilon_{0}\ \frac{ab}{ba}, (a, b = Radius)
 Capacity of cylindrical capacitor \tt C=\frac{2\pi\varepsilon_{0}l}{\log_{e}{\frac{b}{a}}}

Quantity When capacitor is fully charged with air between the two plates When the dielectric slab is introduced without the battery When the dielectric slab is introduced with the Battery Charge Q_{0} Q_{0} Q_{0} Capacity C_{0} KC_{0} KC_{0} P.D between the two plates V_{0} \tt \frac{V_{0}}{K} V_{0} Intensity of electric field E_{0} \tt \frac{E_{0}}{K} E_{0} Enegy stored U_{0} \tt \frac{U_{0}}{K} KU_{0}  If "n" indentical charged liquid drops are combined to form bigdrop

Quantity For ecah charged small drop For the Brg drop Radius r R=n^{1/3} r Charge q Q = nq Capacity c C = n^{1/3} c Potential v V' = n^{2/3} V Energy u U' = n^{5/3} U Surface charge density σ σ' = n^{1/3} σ
View the Topic in this video From 01:36 To 18:00
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1. Work done in charging a capacitor gets stored in the capacitor in the form of its electric potential energy and it is given by
U = \frac{1}{2}CV^{2} = \frac{1}{2}QV = \frac{1}{2}\frac{Q^{2}}{C}
2. Energy Density
u = \frac{1}{2}\varepsilon_{0}E^{2}