Calculating Voltage and Capacitance of a Charged Capacitor

Introduction

The relationship between charge, voltage, and capacitance in a charged capacitor is a fundamental concept in electrical engineering. This article will guide you through the process of calculating the voltage and capacitance of a capacitor that stores a specific amount of energy when charged with a certain amount of charge. We will use two key formulas: the energy stored in a capacitor and the relationship between charge, capacitance, and voltage.

Formulas Used

The two main formulas we will use are:

(E frac{1}{2} C V^2) - Energy stored in the capacitor (Q C V) - Charge on the capacitor

Problem Statement

Given that a capacitor is charged with 10 mC and stores 1.2 J of energy, we need to find the voltage and capacitance of the capacitor.

Step-by-Step Solution

Step 1: Express Capacitance in Terms of Energy and Voltage

From the energy formula (E frac{1}{2} C V^2), we can derive the capacitance in terms of energy and voltage:

C frac{2E}{V^2}

Step 2: Express Voltage in Terms of Charge and Capacitance

From the charge formula (Q C V), we can derive the voltage in terms of charge and capacitance:

V frac{Q}{C}

Step 3: Substitute Voltage from Charge Formula into Energy Formula

Substituting (V frac{Q}{C}) into the energy formula:

E frac{1}{2} C left(frac{Q}{C}right)^2

Which simplifies to:

E frac{1}{2} frac{Q^2}{C}

Step 4: Solving for Capacitance

Rearranging the equation for capacitance:

C frac{Q^2}{2E}

Substituting in the values for charge (Q 10 ; mC 0.01 ; C) and energy (E 1.2 ; J):

C frac{(0.01)^2}{2 times 1.2} frac{0.0001}{2.4} approx 0.00004167 ; F approx 41.67 ; mu F

Step 5: Finding Voltage

Using the formula (V frac{Q}{C}) to find the voltage:

V frac{0.01}{0.00004167} approx 240 ; V

Summary

The capacitance (C) of the capacitor is approximately 41.67 μF and the voltage (V) is approximately 240 V.

Conclusion

We have successfully calculated the voltage and capacitance of a charged capacitor given the energy stored and the charge. Understanding and applying these formulas is crucial for analyzing and designing electrical circuits.