# 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 = U1 + U2 + U3
• In parallel combination of capacitors potential is same
• Total charge is Q = Q1 + Q2 + Q3 + - - - - -
• The Ratio of charges in parallel combination Q1 : Q2 : Q3 = C1 : C2 : C3
• Equivalent capacitance in parallel C = C1 + C2 + C3 + - - - - -
• “n” identical capacitors in parallel Cρ = nc.
• Energy of capacitors U1 : U2 : U3 = C1 : C2 : C3
• Total Energy of combination in parallel U = U1 + U2 + U3
• Two dielectrics K1 , K2 in parallel between capacitors then \tt C_{eff}=\frac{\varepsilon_{0}A}{d}\left(\frac{K_{1}+K_{2}}{2}\right)
• Two dielectrics K1, K2 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 C1 C2 charged to V1 V2 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}{b-a}, (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 Q0 Q0 Q0 Capacity C0 KC0 KC0 P.D between the two plates V0 \tt \frac{V_{0}}{K} V0 Intensity of electric field E0 \tt \frac{E_{0}}{K} E0 Enegy stored U0 \tt \frac{U_{0}}{K} KU0
• If "n" indentical charged liquid drops are combined to form bigdrop
•  Quantity For ecah charged small drop For the Brg drop Radius r R=n1/3 r Charge q Q = nq Capacity c C = n1/3 c Potential v V' = n2/3 V Energy u U' = n5/3 U Surface charge density σ σ' = n1/3 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}