Question

Derive the expression for resultant capacitance, when the capacitor is connected in parallel.

Answer 1

Consider three capacitors of capacitance C₁,C₂ and C₃ connected in parallel with a battery of voltage V as shown in figure (a).

(a) capacitors in parallel-

(b) equivalent capacitance with the same total charge

Since corresponding sides of the capacitors are connected to the same positive and negative terminals of the battery, the voltage across each capacitor is equal to the battery’s voltage. Since capacitance of the capacitors is different, the charge stored in each capacitor is not the same. Let the charge stored in the three capacitors be Q₁,Q₂, and Q₂ respectively. According to the law of conservation of total charge, the sum of these three charges is equal to the charge Q transferred by the battery,

Q = Q₁ + Q₂ + Q₃ ….. (1)

Now, since Q = CV, we have

Q = C₁V + C₂ V + C₃ V ….. (2)

If these three capacitors are considered to form a single capacitance C_P which stores the total charge Q as shown in figure (b), then we can write Q = CPV. Substituting this in equation (2), we get

C_p V = C₁ V + C₂ V + C₃ V

C_p = C₁ + C₂ + C₃

Thus, the equivalent capacitance of capacitors connected in parallel is equal to the sum of the individual capacitance. The equivalent capacitance C_p in a parallel connection is always greater than the largest individual capacitance. In a parallel connection, it is equivalent as area of each capacitance adds to give more effective area such that total capacitance increases.