Capacitors in Combination

Capacitors in combination refer to the arrangement of multiple capacitors in an electric circuit. This arrangement can be either in series or in parallel, each having distinct effects on the overall capacitance and characteristics of the circuit. Series Combination is used to achieve specific voltage ratings or to divide voltage across different components in a circuit. Parallel Combination increases the total capacitance in a circuit, which helps filter noise, stabilize power supplies, and enhance energy storage capacity.

1.0Combination of capacitors 

A combination of capacitors refers to how multiple capacitors are connected within an electric circuit. Capacitors can be arranged in different configurations. Series Combination, Capacitors are connected end-to-end so that the same current flows through each Capacitor. In a parallel combination, capacitors are connected across each other's terminals, so they share the same voltage. Capacitors can be combined in more complex configurations involving series and parallel connections. Calculating total Capacitance in such configurations involves simplifying the circuit by combining capacitors in series or parallel as applicable and then using the respective formulas. These configurations are crucial in circuit design for achieving specific capacitance values and characteristics needed in applications such as energy storage, filtering, and timing circuits. Engineers choose capacitor combinations based on desired capacitors, voltage ratings, physical size constraints, and circuit electrical requirements.

2.0Series Combination of Capacitor

Capacitors are connected end-to-end so that the same current flows through each Capacitor. The total Capacitance in series is less than any individual capacitor's Capacitance. The Charge on each Capacitor connected in series is the same.

The total potential difference V across the arrangement will be divided as V₁, V₂, V₃ across each capacitor C₁, C₂, C₃ respectively.

When a capacitor is connected in series ,the potential drop across the combination is the sum of the potential difference across each capacitor

 V = V₁ + V₂ + V₃

${\mathrm{V}}_1=\frac{Q}{C_1},{\mathrm{V}}_2=\frac{Q}{C_2},{\mathrm{V}}_3=\frac{Q}{C_3}$

$\mathrm{V}=\frac{Q}{C_1}+\frac{Q}{C_2}+\frac{Q}{C_3}$

$\frac{Q}{C}=\frac{Q}{C_1}+\frac{Q}{C_2}+\frac{Q}{C_3}$

$\frac{Q}{C}=\frac{Q}{C_1}+\frac{Q}{C_2}+\frac{Q}{C_3}$

$\frac{1}{C}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}$

$\frac{1}{C}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}+\dots \dots \frac{1}{C_n}\dots$ 

•  When two capacitors are connected in series, than effective capacitance is given by                             

$\frac{1}{C}=\frac{1}{C_1}+\frac{1}{C_2}$

$\frac{1}{C}=\frac{C_1+C_2}{C_1{C}_2}$

$\mathrm{C}=\frac{C_1{C}_2}{C_1+C_2}$

• The Effective capacitance of two capacitors in series is less than that of individual capacitors
• The Charge for each capacitor in the series is the same.

3.0Parallel Combination of Capacitor

Capacitors are connected across each other's terminals and share the same voltage. When a potential difference V is applied across the terminals all capacitors have equal potential difference. The equivalent Capacitance of parallel combination is more significant than any of the capacitances in the combination.

In parallel combination the total charge q is the sum of the charges stored by each capacitor.         

Q = Q₁ + Q₂ + Q₃     (∴ Q₁ = C₁V ,  Q₂ = C₂V,  Q₃ = C₃V)

CV = C₁V + C₂V + C₃V

C=C₁+C₂+C₃

• Effective Capacitance of parallel combination is greater than any of the capacitance
• In Parallel combination, the voltage across each Capacitor is the same.

4.0Electrostatic Energy in a capacitor

Charging a capacitor involves transferring an electric charge between its two plates. This process is fundamental to how capacitors store and release electrical energy.

$\mathrm{dU}=\mathrm{dW}=\mathrm{Vdq}=\frac{q}{C}\mathrm{dq}$ ………….(1)

$\mathrm{U}=\int \mathrm{dU}={\int }_0^Q\frac{q}{C}\mathrm{dq}=\frac{1}{C}{\int }_0^{Q}q\mathrm{dq}=\frac{1}{C}\left[\frac{q^2}{2}\right]=\frac{q^2}{2C}$

$\mathbf{U}=\frac{Q^2}{2C}$ ………….(2)

Put Q = CV in Equation (2)

$\mathbf{U}=\frac{1}{2}{\mathbf{CV}}^2$ ………….(3)

Put $\mathrm{C}=\frac{Q}{V}$ in Equation 3

$\mathbf{U}=\frac{1}{2}\mathbf{QV}$ ………….(4)

5.0Energy Density  of Capacitor

The energy stored in the capacitor resides within the electric field and the space between the capacitor plates. The energy stored per unit volume  occupied between the plates of the Capacitor is known as the Energy Density of the Capacitor.

Energy Density = $\frac{\text{ Energy }}{\text{ Volume }}$

Energy stored in the capacitor is, $\mathrm{U}=\frac{1}{2}{\mathrm{CV}}^2$

The Capacitance of a parallel plate capacitor described as $C=\frac{{\epsilon }_{0}A}{d}$

Potential difference, V = Ed

Energy becomes $\mathrm{U}=\frac{1}{2}{\epsilon }_0{\mathrm{E}}^2\mathrm{Ad}$

The Volume of the capacitor = Ad

Energy Density = $\frac{\text{ Energy }}{\text{ Volume }}=\frac{\frac{1}{2}{\epsilon }_0{E}^{2}Ad}{Ad}=\frac{1}{2}{\epsilon }_0{\mathrm{E}}^2$

$U_E=\frac{1}{2}{\epsilon }_0{\mathrm{E}}^2$    …………. Energy Density of a capacitor

6.0Solved Questions

Q-1. The attractive force between the plates of a parallel plate capacitor is.

Sol.

$F=qE\left(\therefore E=\frac{\sigma }{2{ϵ}_0}\right)$

$\mathrm{F}=\frac{q\sigma }{2{ϵ}_0}\left(\therefore \sigma =\frac{q}{A}\right)$

$\mathrm{F}=\frac{q^2}{2{ϵ}_{0}A}$

Q-2. Two capacitors 0.3F and 0.6F are connected in series across 6 Volts.Find the ratio of energies stored will be.

Sol.

$\frac{C_1}{C_2}=\frac{0.3}{0.6}=\frac{1}{2}$

In series combination, $\frac{U_1}{U_2}=\frac{\frac{Q^2}{2C_1}}{\frac{Q^2}{2C_2}}=\frac{C_2}{C_1}=\frac{2}{1}$

Q-3. Capacitors of 2,3 and 6 F are grouped to give minimum capacitance is.

Sol.

$\frac{1}{C}=\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=\frac{3+2+1}{6}=1\mu F$