Question

1. Using the definition C=Q/V, show that for a SERIES connection of capacitors: 1/C = 1/C1 + 1/C2. What does this equation mean? What happens to C as more and more capacitors are connected in series?
2. Using the definition C=Q/V, show that for a PARALLEL connection of capacitors: C= C1+C2. What does this equation mean? What happens to C as more and more capacitors are connected in parallel?

Answer 1

1) The total voltage is the sum of the individual voltages:

$$V=V_1+v_2+V_3\\\frac{q}{C_s}=\frac{q}{C_1}+\frac{q}{C_2}+\frac{q}{C_3}$$

Canceling the Q_s, we obtain the equation for the total capacitance in series C_S to be

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

An expression of this form always results in a total capacitance C_S that is less than any of the individual capacitances C₁, C₂, …

It tends to zero.

2) Using the relationship Q = CV, we see that the total charge is

$$C_pV=C_1V+C_2V+C_3V$$

Canceling V from the equation, we obtain the equation for the total capacitance in parallel

$$C_p=C_1+C_2+C_3$$

Total capacitance in parallel is simply the sum of the individual capacitances.

It tends to infinity.