Q59PE - College Physics (Urone)

Question

What total capacitances can you make by connecting a $\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\mu F}$ and an $\mathbf{8}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\mu F}$capacitor together?

Step-by-Step Solution

Answer

The equivalent capacitance of the two capacitors in series combination is $3.07 \mathrm{\mu F}$and in parallel combination is $13.0 \mathrm{\mu F}$.

Step 1: Given Data

The capacitances of two capacitors is- $5.00 \mathrm{\mu F}$ and $8.00\mathrm{\mu F}$ .

Step 2: Concept Introduction

Capacitors in Parallel: When capacitors with capacitances  C₁, C₂ ,… are connected in parallel, the equivalent capacitance C_eq equals the sum of the individual capacitances –

Capacitors in Series: When capacitors with capacitances  C₁, C₂, … are connected in series, the reciprocal of the equivalent capacitance C_eq equals the sum of the reciprocals of the individual capacitances –

Step 2: Parallel capacitance

If the two capacitors are connected in parallel combination, their equivalent capacitance is given by Equation (1) as:

 $\begin{array}{rcl}{\mathbf{C}}_{\mathbf{eq}\mathbf{,}\mathbf{p}}& \mathbf{=}& {\mathbf{C}}_{\mathbf{1}}\mathbf{+}{\mathbf{C}}_{\mathbf{2}}\\ & \mathbf{=}& \mathbf{5}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{mF}\mathbf{+}\mathbf{8}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{mF}\\ & \mathbf{=}& \mathbf{13}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{mF}\end{array}$

Step 3: Series Capacitance

If the two capacitors are connected in series combination, their equivalent capacitance is given by Equation (2) :

$\begin{array}{rcl}\frac{\mathbf{1}}{{\mathbf{C}}_{\mathbf{eq}\mathbf{,}\mathbf{s}}}& \mathbf{=}& \frac{\mathbf{1}}{{\mathbf{C}}_{\mathbf{1}}}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{C}}_{\mathbf{2}}}\\ {\mathbf{C}}_{\mathbf{eq}\mathbf{,}\mathbf{s}}& \mathbf{=}& \frac{{\mathbf{C}}_{\mathbf{1}}{\mathbf{C}}_{\mathbf{2}}}{{\mathbf{C}}_{\mathbf{1}}\mathbf{+}{\mathbf{C}}_{\mathbf{2}}}\\ {\mathbf{C}}_{\mathbf{eq}\mathbf{,}\mathbf{s}}& \mathbf{=}& \frac{\mathbf{(}\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\mu F}\mathbf{)}\mathbf{(}\mathbf{8}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\mu F}\mathbf{)}}{\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\mu F}\mathbf{+}\mathbf{8}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\mu F}}\\ & \mathbf{=}& \mathbf{3}\mathbf{.}\mathbf{07}\mathbf{ }\mathbf{\mu F}\end{array}$ 

 

Therefore, equivalent capacitance in the series combination and the parallel combination is $3.07 \mathrm{\mu F}$and $13.0 \mathrm{\mu F}$ respectively.