Combination of Capacitors: Definition, Types, Derivation & Solved Examples

A combination of capacitors is a formation or arrangement of two or more capacitors in an electric circuit that collectively store charge. 

Importance of the Combination of Capacitors for Exams 

• In previous year JEE Mains or related exams like CUET, you will find questions on the combination of capacitors covering how charges are redistributed, how energy gets stored, and how complex combinations work. 
• The combination of capacitors is interrelated to other topics in Class 11 and 12 physics. Some include current electricity, energy stored, and semiconductors. 

By combining capacitors in different ways, we can control how they store and share electric charge. In the world of physics, the combination of capacitors can create differences in charges and voltage. 

So, what are the correct ways to combine them?

We can combine capacitors as 

1. Series Combination 
2. Parallel Combination 
3. Mixed Combination 

When we connect capacitors with the same charge but different potential differences across them, we have a series combination.  

Characteristics of Series Combination

• We can say that ${\mathrm{V}}_1=\frac{\mathrm{Q}}{{\mathrm{C}}_1}$

${\mathrm{V}}_1=$ potential across ${\mathrm{C}}_1$

$\mathrm{Q}=$ charge on positive plate of ${\mathrm{C}}_1$

${\mathrm{C}}_1=$ capacitance of capacitor similarly

${\mathrm{V}}_2=\frac{\mathrm{Q}}{{\mathrm{C}}_2},{\mathrm{V}}_3=\frac{\mathrm{Q}}{{\mathrm{C}}_3};\dots \dots$

• ${\mathrm{V}}_1:{\mathrm{V}}_2:{\mathrm{V}}_3=\frac{1}{{\mathrm{C}}_1}:\frac{1}{{\mathrm{C}}_2}:\frac{1}{{\mathrm{C}}_3}$

The potential difference across capacitor is inversely proportional to its capacitance in series combination.

$\mathrm{V}\propto \frac{1}{\mathrm{C}}$

Note : In series combination, the smallest capacitor gets maximum potential.

• $\mathrm{}{V}_1=\frac{\frac{1}{{\mathrm{C}}_1}}{\frac{1}{{\mathrm{C}}_1}+\frac{1}{{\mathrm{C}}_2}+\frac{1}{{\mathrm{C}}_3}+\dots \dots }\mathrm{V}$ $V_2=\frac{\frac{1}{C_2}}{\frac{1}{{\mathrm{C}}_1}+\frac{1}{{\mathrm{C}}_2}+\frac{1}{{\mathrm{C}}_3}+\dots \dots }\mathrm{V}$ $V_3=\frac{\frac{1}{C_3}}{\frac{1}{{\mathrm{C}}_1}+\frac{1}{{\mathrm{C}}_2}+\frac{1}{{\mathrm{C}}_3}+\dots \dots }\mathrm{V}$ Where $V=V_1+V_2+V_3$
• Equivalent Capacitance : It’s the single capacitor value storing the same charge and energy as the entire combination when used in its place. 

In series : $\frac{1}{{\mathrm{C}}_{\mathrm{e}\mathrm{q}}}=\frac{1}{{\mathrm{C}}_1}+\frac{1}{{\mathrm{C}}_2}+\frac{1}{{\mathrm{C}}_3}+\dots \dots$  

Note : In series combination, the equivalent capacitance is always less the smallest capacitor of the combination.

•  Energy stored in the combination ${\mathrm{U}}_{\text{combin}\text{ation}}=\frac{{\mathrm{Q}}^2}{2{\mathrm{C}}_1}+\frac{{\mathrm{Q}}^2}{2{\mathrm{C}}_2}+\frac{{\mathrm{Q}}^2}{2{\mathrm{C}}_3}$  

${\mathrm{U}}_{\text{combination}}=\frac{{\mathrm{Q}}^2}{2{\mathrm{C}}_{\mathrm{e}\mathrm{q}}}$

Energy the battery supplies in charging the combination $U_{\text{battery}}=\mathrm{Q}\times \mathrm{V}=\mathrm{Q}\cdot \frac{\mathrm{Q}}{{\mathrm{C}}_{\mathrm{e}\mathrm{q}}}=\frac{{\mathrm{Q}}^2}{{\mathrm{C}}_{\mathrm{e}\mathrm{q}}}$

$\frac{U_{\text{combination}}}{U_{\text{battery}}}=\frac{1}{2}$

Note : The battery supplies half of the energy and that is stored as electrostatic energy.  The other half of the energy is converted into heat through resistance (if capacitors are initially uncharged).

$C_{\text{eq}}=\frac{Q}{V}$

Now, initially, the capacitor has no charge.

Applying Kirchoff's Voltage Law - 
$\frac{-\mathrm{Q}}{{\mathrm{C}}_1}+\frac{-\mathrm{Q}}{{\mathrm{C}}_2}+\frac{-\mathrm{Q}}{{\mathrm{C}}_3}+\mathrm{V}=0$

$\mathrm{V}=\mathrm{Q}\left[\frac{1}{{\mathrm{C}}_1}+\frac{1}{{\mathrm{C}}_2}+\frac{1}{{\mathrm{C}}_3}\right];\mathrm{}\mathrm{V}=\frac{1}{{\mathrm{C}}_1}+\frac{1}{{\mathrm{C}}_2}+\frac{1}{{\mathrm{C}}_3}$

$\Rightarrow \mathrm{}\frac{1}{{\mathrm{C}}_{\text{eq}}}=\frac{1}{{\mathrm{C}}_1}+\frac{1}{{\mathrm{C}}_2}+\frac{1}{{\mathrm{C}}_3}$ in general $\frac{1}{{\mathrm{C}}_{\text{eq}}}={\sum }_{\mathrm{n}=1}^{\mathrm{n}} \frac{1}{{\mathrm{C}}_{\mathrm{n}}}$

 

When one plate of each capacitor (more than one) is connected together and the other plate of each capacitor is connected together, such a combination is called a parallel combination.

Characteristics of Parallel Combination of Capacitors

• All capacitors have same potential difference but different charges.
• We can say that :

${\mathrm{Q}}_1={\mathrm{C}}_1\mathrm{V}$
${\mathrm{Q}}_1=$ Charge on capacitor ${\mathrm{C}}_1$
${\mathrm{C}}_1=$ Capacitance of capacitor ${\mathrm{C}}_1$
$\mathrm{V}=$ Potential across capacitor ${\mathrm{C}}_1$

• ${\mathrm{Q}}_1:{\mathrm{Q}}_2:{\mathrm{Q}}_3={\mathrm{C}}_1:{\mathrm{C}}_2:{\mathrm{C}}_3$

The charge on the capacitor is proportional to its capacitance $\mathrm{Q}\propto \mathrm{C}$

• ${\mathrm{Q}}_1=\frac{{\mathrm{C}}_1}{{\mathrm{C}}_1+{\mathrm{C}}_2+{\mathrm{C}}_3}\mathrm{Q}\mathrm{}\Rightarrow \mathrm{}{\mathrm{Q}}_2=\frac{{\mathrm{C}}_2}{{\mathrm{C}}_1+{\mathrm{C}}_2+{\mathrm{C}}_3}\mathrm{Q}$ ${\mathrm{Q}}_3=\frac{{\mathrm{C}}_3}{{\mathrm{C}}_1+{\mathrm{C}}_2+{\mathrm{C}}_3}\mathrm{Q}$
  Where $\mathrm{Q}={\mathrm{Q}}_1+{\mathrm{Q}}_2+{\mathrm{Q}}_3\dots \dots$
  Note: The maximum charge will flow through the capacitor with the largest value.
• Equivalent capacitance of a parallel combination ${\mathrm{C}}_{\text{eq}}={\mathrm{C}}_1+{\mathrm{C}}_2+{\mathrm{C}}_3$

Note that equivalent capacitance is always greater than the largest capacitor of the combination.

•  Energy stored in the combination :
  ${\mathrm{V}}_{\text{combination}}=\frac{1}{2}{\mathrm{C}}_1{\mathrm{V}}^2+\frac{1}{2}{\mathrm{C}}_2{\mathrm{V}}^2+\dots =\frac{1}{2}\left({\mathrm{C}}_1+{\mathrm{C}}_2+{\mathrm{C}}_3\dots ..\right){\mathrm{V}}^2=\frac{1}{2}{\mathrm{C}}_{\text{eq}}{\mathrm{V}}^2$

$U_{\text{battery}}=\mathrm{Q}\mathrm{V}={\mathrm{C}\mathrm{V}}^2$

Here is a derivation for the parallel combination of capacitors. 

The combination which contains the mixing of series parallel combinations or other complex combinations falls in the mixed category. There are two types of mixed combinations

(i) Simple

(ii) Complex.