Unit 11: Electric Circuits – Capacitors, Voltage, and Energy

Electric Circuits: Introduction

Overview

This unit covers the fundamental concepts of electric circuits, focusing on the behavior of charges, voltage, energy changes, and the role of capacitors. The study period is from 16 October to 21 November.

• Electric circuits involve the movement of electric charges through conductors and components.

• Voltage (potential difference) drives the flow of charges, resulting in current and energy transfer.

Electric Potential and Energy

Potential Energy Changes in Electric Fields

When a charge moves through a potential difference, its electric potential energy changes. The largest change occurs when the charge moves across the full voltage drop.

• Electric potential energy ($U$) is the energy a charge possesses due to its position in an electric field.

• The change in potential energy for a charge $q$ moving through a voltage $riangle V$ is: $\triangle U = q \triangle V$

• If a charge starts from rest and is accelerated by the electric field, its final kinetic energy equals the loss in potential energy.

• Example: A proton accelerated across a voltage drop of $2\,\text{V}$ gains kinetic energy $\triangle U = e \times 2\,\text{V}$, where $e$ is the elementary charge.

Capacitors and Voltage

Definition and Function

A capacitor consists of two oppositely charged plates placed parallel to each other, creating a uniform electric field and a fixed voltage drop between them.

• Capacitor: A device that stores electric charge and energy by maintaining a potential difference between its plates.

• Capacitors are used to create uniform fields and store energy efficiently.

• The voltage drop ($\triangle V$) between the plates is proportional to the charge stored.

Capacitor Relationships and Equations

• The work done to move a charge $q$ across a distance $d$ in a uniform field $E$ is: $W = Fd = qEd$

• The voltage between plates: $\triangle V = Ed$

• Surface charge density ($\sigma$): $\sigma = \frac{Q}{A}$

• For parallel plates, the charge is proportional to the voltage: $Q = C \triangle V$

Capacitance (Capacity)

Definition and Formula

Capacitance is the ability of a capacitor to store charge per unit voltage. It is measured in Farads (F).

• For parallel plate capacitors: $C = \varepsilon_0 \frac{A}{d}$ where $\varepsilon_0$ is the permittivity of free space, $A$ is the plate area, and $d$ is the separation distance.

• The charge stored: $Q = \varepsilon_0 \frac{A}{d} \triangle V$

• Capacitance is directly proportional to plate area and inversely proportional to plate separation.

• Example: Increasing the area or decreasing the distance increases capacitance.

Capacitor Design and Uses

Geometry and Manufacturing

• Capacitance increases with larger plate area and decreases with greater plate separation.

• Capacitors are easy to manufacture and come in various shapes and sizes for different applications.

• Applications:

  • Storing charge for later use

  • Maintaining current when batteries fail

  • Smoothing electrical noise and filtering currents

Summary Table: Capacitor Properties

Property	Effect on Capacitance
Plate Area ($A$)	Directly proportional
Plate Separation ($d$)	Inversely proportional
Dielectric Material	Increases capacitance (if $\varepsilon > \varepsilon_0$)

Key Equations

• Change in potential energy: $\triangle U = q \triangle V$

• Capacitance (parallel plates): $C = \varepsilon_0 \frac{A}{d}$

• Charge stored: $Q = C \triangle V$

Additional info:

• Dielectrics (not shown in images but relevant): Inserting a dielectric material between plates increases capacitance by a factor of the material's relative permittivity ($\varepsilon_r$).

• Energy stored in a capacitor: $U = \frac{1}{2} C (\triangle V)^2$