Question

Determine all the values of equivalent capacitance you can create using any combination of three identical capacitors with capacitance \(C\).

Short answer

Answer: The possible equivalent capacitance values are: 1. All capacitors in series: \(C_{eq1} = \frac{C}{3}\) 2. Two capacitors in parallel, and the third one in series: \(C_{eq2} = \frac{2C}{3}\) 3. All capacitors in parallel: \(C_{eq3} = 3C\)

Step-by-step solution

Case 1: All capacitors in series

When all three capacitors are connected in series, the equivalent capacitance C_eq1 can be found using the formula: \(\frac{1}{C_{eq1}} = \frac{1}{C} + \frac{1}{C} + \frac{1}{C}\) Which simplifies to: \(\frac{1}{C_{eq1}} = \frac{3}{C}\) And then: \(C_{eq1} = \frac{C}{3}\)

Case 2: Two capacitors in parallel, and the third one in series

When two capacitors are connected in parallel and the third one is connected in series with this pair, the equivalent capacitance C_eq2 can be found using the following steps: 1. Find the equivalent capacitance of the two capacitors in parallel (C_12). The formula for the equivalent capacitance of capacitors in parallel is: \(C_{12} = C + C = 2C\) 2. Now, find the equivalent capacitance of C_12 and the third capacitor C connected in series. Using the formula for the equivalent capacitance of capacitors in series: \(\frac{1}{C_{eq2}} = \frac{1}{C_{12}} + \frac{1}{C}\) Substitute the value of C_12: \(\frac{1}{C_{eq2}} = \frac{1}{2C} + \frac{1}{C}\) Simplifying, we get: \(\frac{1}{C_{eq2}} = \frac{3}{2C}\) And then: \(C_{eq2} = \frac{2C}{3}\)

Case 3: All capacitors in parallel

When all three capacitors are connected in parallel, the equivalent capacitance C_eq3 can be found using the formula: \(C_{eq3} = C + C + C = 3C\)

Summary

There are three possible equivalents capacitances that can be created using any combination of three identical capacitors with capacitance C: 1. All capacitors in series: \(C_{eq1} = \frac{C}{3}\) 2. Two capacitors in parallel, and the third one in series: \(C_{eq2} = \frac{2C}{3}\) 3. All capacitors in parallel: \(C_{eq3} = 3C\) These are all the possible values of equivalent capacitance that can be created using three identical capacitors with capacitance C.

Key concepts

Series and Parallel Circuits

In electrical circuits, capacitors can be arranged in series or parallel configurations.
Knowing how to calculate the equivalent capacitance in these setups is crucial for understanding how circuits function.
Let's break down what these terms mean and how they differ:

• Series Circuits: Capacitors are said to be in series when they are connected one after the other on the same path. In such arrangements, the total (or equivalent) capacitance is reduced. The reason is that the electric field has to be established across all the capacitors simultaneously. Mathematically, if three capacitors each of capacitance \( C \) are in series, the equivalent capacitance \( C_{eq} \) is calculated using: \( \frac{1}{C_{eq}} = \frac{1}{C} + \frac{1}{C} + \frac{1}{C} \), which simplifies to \( C_{eq} = \frac{C}{3} \).
• Parallel Circuits: Capacitors in parallel share the same two endpoints, meaning each capacitor is connected directly across the voltage source. This layout increases the equivalent capacitance because each capacitor adds its charge-holding capacity to the overall circuit. For three capacitors in parallel, the capacitance \( C_{eq} \) is simply \( C + C + C = 3C \).

By understanding these two basic configurations, you can figure out different ways to manipulate circuit behavior and design circuits that fit specific needs.

Equivalent Capacitance

The concept of equivalent capacitance is essential in circuit analysis. It helps simplify and reduce a complex circuit into a single, equivalent capacitor that has the same effect on the circuit's charging and discharging properties.
Let's delve into how this works with some examples:

• When capacitors are connected in series, the equivalent capacitance \( C_{eq} \) is lower than each individual capacitance. This occurs because the total separation between the collective plates of the capacitors increases, thereby decreasing the overall capacitance.
• When capacitors are in parallel, each one supplements the total capacitance. As more capacitors are added in parallel, the total capacitance \( C_{eq} \) increases, effectively increasing the circuit's ability to store charge.

For example:

• All capacitors in series: \( C_{eq} = \frac{C}{3} \).
• Two capacitors in parallel and the third in series, which combines both configurations: \( C_{eq} = \frac{2C}{3} \).
• All capacitors in parallel: \( C_{eq} = 3C \).

Understanding these configurations helps in designing circuits for specific functions, like tuning the response time or controlling the flow of electricity.

Electrical Circuits

Electrical circuits consist of various components that work together to perform a function.
Capacitors are one such component that plays a vital role in many circuits, especially when it comes to storing and releasing energy.

• Role of Capacitors: They temporarily store electrical energy when connected to a power source, and can release this energy when required. This ability is crucial in applications like timing circuits, filtering signals, and managing power supply interruptions.
• Capacitors in Circuits: They are often used in both series and parallel arrangements within circuits for different purposes. Knowing how to configure them changes the total capacitance and therefore alters the circuit's behavior. For instance, reducing capacitance via a series setup might be useful for applications needing rapid discharge or charge control.

Basics of how capacitors interact within circuits allows for precise control over electronic devices and systems.
From small gadgets to large industrial machines, capacitors are essential for managing how and when electricity is used.