Lesson 6.3: Capacitance

Introduction

Welcome to Lesson 6.3 of Foundation Physics! In this lesson, we will explore the concept of capacitance. By the end of this lesson, students, you will understand how capacitors work, their energy storage capabilities, and the importance of the time constant in charging and discharging processes.

Learning Objectives:

• Understand capacitance and the farad as a unit of measure.
• Describe the structure and function of a parallel-plate capacitor.
• Calculate the energy stored in a capacitor using the formulas $\frac{1}{2}QV$ and $\frac{1}{2}CV^2$.
• Discuss the charging and discharging of a capacitor through a resistor and the application of the time constant $RC$.
• Analyze exponential decay in charge, current, and voltage.
• Calculate capacitance, charge, and stored energy effectively.

What is Capacitance?

Capacitance is the ability of a system to store an electric charge. This property is quantified in farads (F), named after the scientist Michael Faraday. A capacitor, which is an electrical component that stores energy, is the device that has capacitance!

The Parallel-Plate Capacitor

A parallel-plate capacitor consists of two conductive plates separated by a dielectric material (an insulator). When a voltage $V$ is applied across the plates, positive charge accumulates on one plate, while an equal amount of negative charge accumulates on the other plate. The capacitance $C$ of a parallel-plate capacitor is given by:

$$ C = \frac{\varepsilon_0 A}{d} $$

where:

• $C$ is the capacitance in farads (F),
• $\varepsilon_0$ is the permittivity of free space ($8.85 \times 10^{-12} F/m$),
• $A$ is the area of one of the plates in square meters (m²), and
• $d$ is the separation between the plates in meters (m).

Energy Stored in a Capacitor

When a capacitor is charged, it stores energy. The energy ($U$) stored in a capacitor can be expressed in two equivalent formulas:

$$ U = \frac{1}{2}QV $$

$$ U = \frac{1}{2}CV^2 $$

where:

• $U$ is the energy in joules (J),
• $Q$ is the charge stored in coulombs (C), and
• $V$ is the voltage across the capacitor in volts (V).

Charging and Discharging a Capacitor

When a capacitor is connected to a battery, it charges until the voltage across it equals the battery voltage. The charging process through a resistance $R$ can be expressed by the formula:

$$ V(t) = V_0(1 - e^{-\frac{t}{RC}}) $$

where:

• $V_0$ is the maximum voltage,
• $t$ is the time in seconds (s), and
• $e$ is Euler’s number (approximately 2.718).

The time constant $\tau = RC$ determines how quickly the capacitor charges. A larger time constant means a slower charge time. Similarly, when discharging, the voltage drops according to:

$$ V(t) = V_0e^{-\frac{t}{RC}} $$

Exponential Decay

In both charging and discharging, the behavior of the voltage, current, and charge shows exponential characteristics. For example, the current $I(t)$ during charging can be described as:

$$ I(t) = \frac{V_0}{R} e^{-\frac{t}{RC}} $$

This indicates that the current decreases exponentially over time until it approaches zero as the capacitor becomes fully charged.

Similarly, during discharge, the current decreases and can be constrained by:

$$ I(t) = -\frac{Q_0}{RC} e^{-\frac{t}{RC}} $$

Conclusion

In this lesson, we have explored the fundamental concepts of capacitance, parallel-plate capacitors, energy storage, and the charging and discharging processes. Understanding these principles forms a cornerstone in the study of electric fields and electromagnetism.

Study Notes

• Capacitance is measured in farads (F).
• A typical configuration for a capacitor is a parallel-plate design.
• The formulas for energy stored in a capacitor are:
• $U = \frac{1}{2}QV$
• $U = \frac{1}{2}CV^2$
• The charging formula for a capacitor: $V(t) = V_0(1 - e^{-\frac{t}{RC}})$
• The discharging formula for a capacitor: $V(t) = V_0e^{-\frac{t}{RC}}$
• The time constant ($\tau = RC$) is significant in determining charge/discharge rates.