Electrostatics :: Problem 8

Problem 8: Find the equivalent capacitance of the combination shown in the following diagram.
Solution:

Step: 1 Overview

Parallel combination of capacitors:
When capacitors are connected in parallel, the voltage drop across each capacitor will be same. While charge will be divided across each capacitor according to its capacitance.
Suppose three capacitors are connected in parallel, with capacitances C₁, C₂ and C₃. Total charge will be:

Q = Q₁+Q₂+Q₃

Since Q=CV,

CV = C₁V₁+C₂V₂+C₃V₃

we know that in parallel combination voltage across each capacitor would be same, i,e; V₁=V₂=V₃=V

CV = C₁V+C₂V+C₃V
CV = (C₁+C₂+C₃)V
C = C₁+C₂+C₃

From above equation we can conclude that total capacitance in parallel combination is simply an arithmetic sum of all the capacitances of capacitors connected in parallel.

Series combination of capacitors:
When capacitors are connected in series, each capacitor stores same amount of charge, while total voltage is divided across each capacitor.
Suppose three capacitors are connected in parallel, with capacitances C₁, C₂ and C₃. Total voltage will be:

V = V₁ + V₂ + V₃

Since Q=CV, V=Q/C,

Q/C = Q₁/C₁ + Q₂/C₂ + Q₃/C₃

In series, each capacitor stores same amount of charge, so Q₁=Q₂=Q₃=Q

Q/C = Q/C₁ + Q/C₂ + Q/C₃
Q/C = Q(1/C₁ + 1/C₂ + 1/C₃)
1/C = 1/C₁ + 1/C₂ + 1/C₃

So in this combination reciprocal of total capacitance is the sum of reciprocals of all the capacitances of capacitors connected in series.

Step: 2 Calculation

Since above three highlighted capacitors are in series, so their equivalent capacitance (C_x) will be:

1/C_x = 1/3 + 1/3 +1/3
1/C_x = 3/3 = 1
C_x = 1μf

After substituting the 1μf instead of three series capacitors, we can see that now 1μf and 2μf are in parallel, so their equivalent (C_y) will be:

C_y = 1 + 2
C_y = 3μf

Now after further simplification, again three 3μf capacitors are in series, So their equivalent(C_z) will be:

1/C_z = 1/3 + 1/3 +1/3
1/C_z = 3/3 = 1
C_z = 1μf

Again combination of series capacitors (1μf) and 2μf are in parallel, so their equivalent(C_t) will be:

C_t = 1 + 2
C_t =3μf

Now finally three capacitors are series so their equivalent will be:

1/C = 1/3 + 1/3 +1/3
1/C = 3/3 = 1
C = 1μf (Ans)