JEE Main & Advanced Physics Electrostatics & Capacitance Grouping of Capacitor

(1) Series grouping

(i) Charge on each capacitor remains same and equals to the main charge supplied by the battery but potential difference distributes  i.e. \[V={{V}_{1}}+{{V}_{2}}+{{V}_{3}}\]

(ii) Equivalent capacitance

\[\frac{1}{{{C}_{eq}}}=\frac{1}{{{C}_{1}}}+\frac{1}{{{C}_{2}}}+\frac{1}{{{C}_{3}}}\] or \[{{C}_{eq}}={{(C_{1}^{-1}+C_{2}^{-1}+C_{3}^{-1})}^{-1}}\]

(iii) In series combination potential difference and energy distributes in the reverse ratio of capacitance i.e.,

\[V\propto \frac{1}{C}\] and \[U\propto \frac{1}{C}\].

(iv) If two capacitors having capacitances \[{{C}_{1}}\] and \[{{C}_{2}}\] are connected in series then \[{{C}_{eq}}=\frac{{{C}_{\mathbf{1}}}{{C}_{\mathbf{2}}}}{{{C}_{\mathbf{1}}}+{{C}_{\mathbf{2}}}}=\frac{Multiplication}{Addition}\]

\[{{V}_{1}}=\left( \frac{{{C}_{2}}}{{{C}_{1}}+{{C}_{2}}} \right)\,.\,V\]  and \[{{V}_{2}}=\left( \frac{{{C}_{1}}}{{{C}_{1}}+{{C}_{2}}} \right)\,.\,V\]

(v) If \[n\] identical capacitors each having capacitances C are connected in series with supply voltage V then Equivalent capacitance \[{{C}_{eq}}=\frac{C}{n}\,\] and Potential difference across each capacitor \[V'=\frac{V}{n}\].

(vi) If \[n\] identical plates are arranged as shown below, they constitute \[(n-1)\] capacitors in series. If each capacitors having capacitance \[\frac{{{\varepsilon }_{0}}A}{d}\] then  \[{{C}_{eq}}=\frac{{{\varepsilon }_{0}}A}{(n-1)d}\]

In this situation except two extreme plates each plate is common to adjacent capacitors.

(2) Parallel grouping

(i) Potential difference across each capacitor remains same and equal to the applied potential difference but charge distributes i.e. \[Q={{Q}_{1}}+{{Q}_{2}}+{{Q}_{3}}\]

(ii) \[{{C}_{-eq}}={{C}_{1}}+{{C}_{2}}+{{C}_{3}}\]

(iii) In parallel combination charge and energy distributes in the ratio of capacitance i.e. \[Q\,\,\propto \,\,C\] and \[U\,\,\propto \,\,C\]

(iv) If two capacitors having capacitance \[{{C}_{1}}\] and \[{{C}_{2}}\] respectively are connected in parallel then \[{{C}_{eq}}={{C}_{1}}+{{C}_{2}}\]

\[{{Q}_{1}}=\left( \frac{{{C}_{1}}}{{{C}_{1}}+{{C}_{2}}} \right)\,.\,Q\] and  \[{{Q}_{2}}=\,\left( \frac{{{C}_{2}}}{{{C}_{1}}+{{C}_{2}}} \right)\,.\,Q\]

(v) If \[n\] identical capacitors are connected in parallel Equivalent capacitance \[{{C}_{eq}}=nC\] and Charge on each capacitor \[Q'=\frac{Q}{n}\]

If \[n\] identical plates are arranged such that even numbered of plates are connected together and odd numbered plates are connected together, then \[(n-1)\]  capacitors will be formed and they will be in parallel grouping.

Equivalent capacitance \[C'=(n-1)\,C\] where \[C=\] capacitance of a capacitor \[=\frac{{{\varepsilon }_{0}}A}{d}\]