Capacitors in Parallel

Capacitors in parallel refer to the capacitors that are connected together in parallel when the connection of both of its terminals takes place to each terminal of another capacitor. Furthermore, the voltage’s ( Vc ) connected across all the capacitors, whose connection is in parallel, is the same. Then, capacitors in parallel across them have a “common voltage” supply.

Introduction to Capacitors in Parallel

Multiple connections of capacitors fulfil the role of a single equivalent capacitor. Furthermore, the total capacitance of this equivalent single capacitor is dependent on both the individual capacitors and how their connection takes place. Moreover, there are two simple types of connections- series and parallel, which facilitate the easy calculation of the total capacitance.

Capacitors may be placed in parallel as they provide higher levels of capacitance. Furthermore, capacitors in parallel give us a distributed capacitance on a printed circuit board. Moreover, they give us an exact value which may not have been available otherwise.

Working of Capacitors in Parallel

The design of a capacitor is such that it helps in storing the energy in the form of the electric field, electrostatic energy. Whenever a need arises to increase more electrostatic energy storing capacity, there would be a requirement for a suitable capacitor of increased capacitance. A capacitor involves two metal plates whose connection takes place in parallel and their separation is by a dielectric medium like ceramics, glass, mica etc.

There would be a non-conducting medium between the plates due to the dielectric. Furthermore, the dielectric is characterized by a special ability to hold the charge. Moreover, the capacitance of the capacitor refers to the ability of the capacitor to store charge.

Formula of Capacitors in Parallel

Below is the capacitors in parallel formula:

The formula, Ceq = C1 + C2 + C3 +……+ Cn

Derivation of the Formula of Capacitors in Parallel

When the connection of a voltage source takes place across the plates of the capacitor such that there is a positive charge on one plate, the other plate’s negative charge will be deposited. The total amount of charge (q) whose accumulation takes place, is directly proportional to the voltage source (V) such that,

q = CV      (1)

Where C refers to the proportionality constant i.e. capacitance. Furthermore, its value is dependent on the physical dimensions of the capacitor.

Mathematically, C = \(\frac{\epsilon A}{d}\)

Where A = effective plate area, d = space between plates, and ε = dielectric constant.

When the connection of two capacitors takes place in parallel, then the voltage (V) across each capacitor would be the same i.e. (V_eq = V_a = V_b) and the division of current ( i_eq ) takes place into two parts i_a and i_b. As one knows that

i = \(\frac{dq}{dt}\)

Putting, in the above equation,  the value of q from equation (1),

i = \(\frac{d\left ( CV \right )}{dt} \Rightarrow i = C\frac{dV}{dt} + V\frac{dC}{dt}\)

The later term would be zero (as capacitor’ capacitance is constant). Therefore,

i = \(C\frac{dV}{dt}\)

Applying Kirchhoff’s Current Law at the parallel connection’s  incoming node

i_eq = i_a + i_b

i_eq = \(C_{a}\frac{dV_{a}}{dt} + C_{b}\frac{dV_{b}}{dt}\)

V_eq = V_a = V_b

Therefore, i_eq = \(C_{a}\frac{dV_{eq}}{dt} + C_{b}\frac{dV_{eq}}{dt} \Rightarrow i_{eq} = \left ( C_{a}+C_{b} \right )\frac{dV_{eq}}{dt}\)

Finally we get,

i_eq = \(C_{eq}\frac{dV_{eq}}{dt}\)

Where, C_eq = C_a + C_b

Hence, whenever the connection of n capacitors takes place in parallel, the equivalent capacitance of the whole connection can be expressed by following equation. Furthermore, this equation is similar to the equivalent resistance of resistors when the connection takes place in series.

C_eq = C₁ + C₂ + C₃ + ….+ C_n

FAQs For Capacitors in Parallel

Question 1: Explain the charge on capacitors in parallel?

Answer 1: Consider a case where the voltage across the capacitors is the same as that across the voltage source. As such, the charge on capacitors in parallel will be the same on them as it would have been on if they were connected individually to the voltage source.

Question 2: What is the equivalent capacitance in parallel?

Answer 2: When the connection of several capacitors takes place in a parallel combination, the sum of the individual capacitances will be the equivalent capacitance.