Electric Charge and Electric Fields

Summary

This chapter introduces electrostatics, the study of electric charges at rest and the forces they exert. You'll discover that charge comes in two types—positive and negative—and is conserved in all processes. Materials are classified as conductors, insulators, semiconductors, or superconductors based on how easily charge flows through them. Charge can be transferred through friction, contact, or induction, and grounding provides a path for excess charge to dissipate. Coulomb's Law quantifies the electric force between charges, analogous to Newton's law of gravitation. This force can be visualized using the electric field concept, represented by electric field lines and characterized by field strength. Moving charges in electric fields involves electric potential energy, which leads to the concepts of electric potential and voltage—the energy per unit charge. These foundational concepts in electricity underlie all electronic technology and prepare you for deeper study of circuits and electromagnetism.

Concepts Covered

This chapter covers the following 20 concepts from the learning graph:

1. Electric Charge
2. Positive Charge
3. Negative Charge
4. Conservation of Charge
5. Conductors
6. Insulators
7. Semiconductors
8. Superconductors
9. Charging by Friction
10. Charging by Contact
11. Charging by Induction
12. Grounding
13. Coulomb's Law
14. Electric Force
15. Electric Field
16. Electric Field Lines
17. Field Strength
18. Electric Potential Energy
19. Electric Potential
20. Voltage

Prerequisites

This chapter builds on concepts from:

• Chapter 1: Scientific Foundations and Mathematical Tools
• Chapter 4: Forces and Newton's Laws
• Chapter 6: Work, Energy, and Power

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Introduction to Electric Charge

Have you ever felt a shock when touching a doorknob after walking across a carpet? Or noticed how your hair stands on end when you rub a balloon against it? These everyday experiences reveal one of the most fundamental forces in nature: electricity. Unlike gravity, which only attracts, electric forces can both attract and repel. This chapter explores the nature of electric charge, how it moves through different materials, and the forces and fields that govern its behavior.

The Nature of Electric Charge

What Is Electric Charge?

Electric charge is a fundamental property of matter, much like mass. Just as mass determines how objects respond to gravitational forces, charge determines how objects respond to electric forces. However, unlike mass—which is always positive—electric charge comes in two varieties: positive and negative.

Positive charge is associated with protons in atomic nuclei, while negative charge is associated with electrons orbiting those nuclei. These names are historical conventions chosen by Benjamin Franklin, who didn't know about subatomic particles but recognized that electricity came in two types that behaved differently.

Key properties of electric charge:

• Charge is quantized (comes in discrete units)
• The fundamental unit of charge is the electron charge: e = 1.6 × 10^-19^ coulombs (C)
• Opposite charges attract; like charges repel
• Charge can be transferred but never created or destroyed

Conservation of Charge

One of the most important principles in all of physics is the conservation of charge. This law states that the total electric charge in an isolated system remains constant. Charge can move from one object to another, but it cannot be created or destroyed.

When you rub a balloon on your hair, electrons transfer from your hair to the balloon. Your hair loses electrons (becoming positively charged), while the balloon gains the same number of electrons (becoming negatively charged). The total charge of the system—you plus the balloon—hasn't changed; it's just been redistributed.

Diagram: Charge Conservation in Chemical Reactions

<summary>Charge Conservation in Chemical Reactions</summary>
Type: diagram

Purpose: Illustrate how charge is conserved in a simple chemical reaction

Components to show:
- Before reaction: Na atom (neutral, 11 protons, 11 electrons) and Cl atom (neutral, 17 protons, 17 electrons)
- Electron transfer: Arrow showing one electron moving from Na to Cl
- After reaction: Na+ ion (11 protons, 10 electrons, +1 charge) and Cl- ion (17 protons, 18 electrons, -1 charge)
- Charge tally boxes showing total charge before (0) equals total charge after (0)

Connections:
- Electron transfer arrow between Na and Cl atoms
- Plus and minus symbols on resulting ions

Style: Atomic diagrams with nucleus and electron shells, colorful and engaging

Labels:
- "Sodium (Na)" and "Chlorine (Cl)" atoms
- "Electron transfer"
- "Total charge before: 0"
- "Total charge after: +1 + (-1) = 0"

Color scheme: Blue for atoms/ions, gold for electron in motion, green for charge tally boxes

Electrical Properties of Materials

Materials respond very differently to electric forces. Some materials allow charge to flow freely through them, while others block its movement almost completely. Understanding these differences is crucial for everything from designing electronic circuits to preventing lightning strikes.

Conductors

Conductors are materials that allow electric charge to flow freely through them. In conductors, some electrons are loosely bound to their atoms and can move throughout the material. When you apply an electric force to one end of a conductor, these free electrons respond immediately, moving through the entire material.

Common conductors include:

• Metals (copper, aluminum, silver, gold)
• Graphite (a form of carbon)
• Saltwater and other ionic solutions
• The human body (due to its water content)

In metals, the atomic structure creates a "sea of electrons" that can flow freely. This is why electrical wires are made of copper or aluminum—these materials offer very little resistance to the flow of charge.

Insulators

Insulators are materials that resist the flow of electric charge. In these materials, electrons are tightly bound to their atoms and cannot move freely. Even when you apply a strong electric force, the charges stay put (or move only tiny distances).

Common insulators include:

• Rubber
• Plastic
• Glass
• Ceramic
• Dry air
• Most pure (distilled) water

Insulators are essential for electrical safety. The plastic coating on electrical wires prevents charges from leaking out. The rubber soles on your shoes protect you from electrical shocks. Even the air around us acts as an insulator—until the electric force becomes so strong that it ionizes the air molecules, creating a conducting path (lightning).

Semiconductors and Superconductors

Between the extremes of conductors and insulators lie two special categories of materials.

Semiconductors are materials that conduct electricity better than insulators but not as well as conductors. More importantly, their conductivity can be controlled by adding impurities (doping), applying voltage, or changing temperature. Silicon and germanium are the most important semiconductors.

Semiconductors form the basis of all modern electronics:

• Computer processors contain billions of semiconductor transistors
• Solar cells convert light to electricity using semiconductor materials
• LED lights produce light through semiconductor junctions
• Smartphone sensors detect light, sound, and motion using semiconductors

Superconductors are materials that, below a certain critical temperature, conduct electricity with exactly zero resistance. In normal conductors, moving electrons collide with atoms, generating heat and losing energy. In superconductors, electrons pair up and move through the material without any resistance at all.

Applications of superconductors:

• MRI machines use superconducting magnets
• Particle accelerators like the Large Hadron Collider
• Maglev trains that float using magnetic levitation
• Ultra-sensitive magnetic field detectors

The challenge with superconductors is that most require extremely low temperatures (near absolute zero) to work. Scientists are actively searching for "high-temperature" superconductors that work at more practical temperatures.

Material Type	Conductivity	Key Characteristic	Example
Conductor	High	Free electrons	Copper wire
Insulator	Very low	Electrons tightly bound	Rubber
Semiconductor	Variable	Controllable conductivity	Silicon
Superconductor	Infinite (below T~c~)	Zero resistance	Niobium-titanium

Methods of Charging Objects

How do neutral objects become charged? There are three main methods for transferring charge from one object to another: friction, contact, and induction.

Charging by Friction

Charging by friction occurs when two different materials are rubbed together. The friction causes electrons to transfer from one material to the other, leaving one object with a surplus of electrons (negatively charged) and the other with a deficit (positively charged).

Classic examples:

• Rubbing a balloon on hair (balloon becomes negative, hair positive)
• Shuffling feet on carpet (shoes gain electrons, carpet loses them)
• Rubbing a plastic rod with wool (rod becomes negative, wool positive)
• Rubbing a glass rod with silk (rod becomes positive, silk negative)

The direction of electron transfer depends on each material's relative affinity for electrons. The triboelectric series ranks materials by their tendency to give up or accept electrons. Materials higher on the series (like rabbit fur) tend to lose electrons, while materials lower on the series (like PVC plastic) tend to gain electrons.

Diagram: Triboelectric Series Interactive Infographic

<summary>Triboelectric Series Interactive Infographic</summary>
Type: infographic

Purpose: Show the triboelectric series and demonstrate which materials become positive or negative when rubbed together

Layout: Vertical gradient bar from most positive (top) to most negative (bottom)

Materials listed (top to bottom):
- Rabbit fur (most positive)
- Glass
- Human hair
- Nylon
- Wool
- Silk
- Paper
- Cotton
- Wood
- Amber
- Rubber balloon
- Polyester
- PVC (most negative)

Interactive elements:
- Hover over any material to see its position highlighted
- Click two materials to see which becomes positive/negative when rubbed together
- Arrow appears showing direction of electron transfer
- Color coding: selected materials glow (positive = red, negative = blue)

Visual style: Vertical gradient bar transitioning from red (positive) through neutral (white) to blue (negative)

Annotations:
- "More positive (electron donors)" at top
- "More negative (electron acceptors)" at bottom
- "When two materials are rubbed together, electrons flow from the higher material to the lower material"

Implementation: HTML/CSS/JavaScript with interactive click handlers

Charging by Contact

Charging by contact occurs when a charged object touches a neutral object, allowing charge to flow directly from one to the other. Unlike friction charging, which requires rubbing, contact charging only requires physical touch.

How it works:

1. A charged object touches a neutral conductor
2. Charges redistribute between the objects
3. Both objects end up with the same type of charge (both positive or both negative)
4. The charge is shared, so each object ends up with less charge than the original

For example, if a negatively charged rod touches a neutral metal sphere, some electrons flow from the rod to the sphere. Now both objects are negatively charged, though the rod has less negative charge than before.

This method is commonly used to charge electroscopes (devices that detect electric charge) and to transfer static charge in various applications.

Charging by Induction

Charging by induction is a clever method that allows you to charge an object without even touching it. This method uses the influence of a nearby charged object to redistribute charge, then removes part of the object to leave it with a net charge.

The process:

1. Bring a charged object near (but not touching) a neutral conductor
2. The nearby charge causes electrons in the conductor to move, creating regions of opposite charge
3. Ground one side of the conductor (allowing charge to flow to or from Earth)
4. Remove the ground connection, trapping the separated charges
5. Remove the original charged object, leaving the conductor with a net charge

The remarkable feature of induction is that the conductor ends up with the opposite charge from the nearby charged object, even though they never touched. A negatively charged rod can induce a positive charge in a conductor, and vice versa.

Diagram: Charging by Induction Animation MicroSim

<summary>Charging by Induction Animation MicroSim</summary>
Type: microsim

Learning objective: Demonstrate the step-by-step process of charging a neutral conductor by induction

Canvas layout (800x600px):
- Main area (600x600): Animation showing the induction process
- Control panel (200x600): Step controls and explanations

Visual elements:
- Negatively charged rod (blue with minus signs)
- Neutral metal sphere on insulating stand (gray initially)
- Ground connection (wire with Earth symbol)
- Free electrons shown as small blue dots
- Positive charges (nucleus regions) shown as red plus signs
- Field lines showing electric influence (dashed yellow lines)

Interactive controls:
- Button: "Next Step" / "Previous Step"
- Button: "Reset"
- Checkbox: "Show electron movement"
- Checkbox: "Show field lines"
- Text display: Current step explanation

Animation steps:
1. Initial state: Rod far away, sphere neutral, electrons evenly distributed
2. Rod approaches: Electrons in sphere move away from rod (polarization)
3. Ground connected: Electrons flow to ground, leaving sphere electron-deficient on near side
4. Ground disconnected: Sphere now has trapped positive charge
5. Rod removed: Positive charge redistributes evenly over sphere

Default parameters:
- Animation speed: 1 second per step
- Show electron movement: ON
- Show field lines: OFF

Behavior:
- Electrons animated moving smoothly during redistribution
- Color change of sphere from gray (neutral) to pink (positive)
- Step explanations update in control panel
- Can jump to any step or play through automatically

Implementation notes:
- Use p5.js for rendering
- Draw electrons as small circles, animate with lerp() for smooth motion
- Use alpha transparency for field lines
- Color code: blue for negative, red/pink for positive, gray for neutral

Grounding

Grounding (also called earthing) is the process of connecting an object to the Earth, which acts as an infinite reservoir of charge. When you ground an object, excess charge flows between the object and Earth until the object reaches the same electric potential as the ground.

Why grounding matters:

• Safety: Lightning rods ground buildings to safely conduct lightning strikes to Earth
• Electronics: Computer cases are grounded to prevent static discharge damage
• Static discharge: Touch a grounded metal object before handling sensitive electronics
• Charge removal: Grounding removes unwanted static charge from objects

In the process of charging by induction, grounding provides a path for charges to flow away, leaving the object with a net charge opposite to the inducing charge.

Coulomb's Law: Quantifying Electric Force

Now that we understand what electric charge is and how objects become charged, we can explore the forces between charged objects. The French physicist Charles-Augustin de Coulomb discovered the mathematical relationship governing these forces in the 1780s.

The Electric Force

Electric force is the push or pull that charged objects exert on each other. Like gravitational force, electric force acts at a distance—charged objects don't need to touch to feel the force. However, unlike gravity (which only attracts), electric force can both attract and repel.

Key properties of electric force:

• Opposite charges attract (positive-negative)
• Like charges repel (positive-positive or negative-negative)
• Force strength increases with larger charges
• Force strength decreases with greater distance
• Force acts along the line connecting the two charges

Coulomb's Law Formula

Coulomb's Law gives the magnitude of the electric force between two point charges:

$$F = k \frac{|q_1 q_2|}{r^2}$$

where:

• F = magnitude of electric force (in newtons, N)
• k = Coulomb's constant = 8.99 × 10^9^ N·m^2^/C^2^
• q~1~ and q~2~ = magnitudes of the two charges (in coulombs, C)
• r = distance between the charges (in meters, m)

Notice the similarity to Newton's law of gravitation:

$$F_g = G \frac{m_1 m_2}{r^2}$$

Both forces follow an inverse square law—doubling the distance reduces the force to one-quarter its original value. However, there's a crucial difference: gravitational force is always attractive, while electric force can be attractive or repulsive.

Comparing Electric and Gravitational Forces

How do electric and gravitational forces compare in strength? Consider two electrons:

• Electric repulsion: F~e~ ≈ 2.3 × 10^-8^ N
• Gravitational attraction: F~g~ ≈ 5.5 × 10^-71^ N
• Ratio: F~e~/F~g~ ≈ 4.2 × 10^42^

The electric force is about 10^42^ times stronger than the gravitational force! This enormous difference explains why electric forces dominate at atomic scales, determining the structure of atoms, molecules, and materials. Gravity only becomes significant for large objects, where the huge number of atoms provides enough mass to generate noticeable gravitational effects.

Diagram: Coulomb's Law Force Calculator MicroSim

<summary>Coulomb's Law Force Calculator MicroSim</summary>
Type: microsim

Learning objective: Allow students to explore how electric force depends on charge magnitudes and separation distance

Canvas layout (900x600px):
- Visualization area (600x600): Shows two charged spheres and force vectors
- Control panel (300x600): Input sliders and force display

Visual elements:
- Two charged spheres (color coded by charge sign: blue = negative, red = positive)
- Distance line between sphere centers (with measurement)
- Force vectors as arrows (green for attractive, red for repulsive)
- Vector length proportional to force magnitude (with scaling)
- Background grid for reference

Interactive controls:
- Slider: q₁ charge (-10 to +10 μC)
- Slider: q₂ charge (-10 to +10 μC)
- Slider: distance (0.5 to 5.0 m)
- Display: Force magnitude (in newtons, with scientific notation)
- Display: Force direction (attractive or repulsive)
- Button: "Randomize charges"
- Button: "Reset to defaults"
- Checkbox: "Show inverse square relationship graph"

Default parameters:
- q₁ = +5 μC
- q₂ = -5 μC
- distance = 1.0 m
- Show graph: OFF

Behavior:
- Spheres move apart or closer as distance slider changes
- Sphere colors change based on charge sign
- Force vectors update in real-time as parameters change
- Force magnitude display updates with color coding (green if < 1 N, yellow if 1-10 N, red if > 10 N)
- Optional graph shows F vs. r curve for current charge values

Additional features:
- Comparison mode: Show gravitational force between equal masses for comparison
- Warning message if charges get too close (unrealistic forces)
- Annotation showing 1/r² relationship when distance changes

Implementation notes:
- Use p5.js for rendering
- Coulomb constant k = 8.99 × 10⁹ N·m²/C²
- Vector arrow scaling: Use logarithmic scale for large force ranges
- Graph: Plot F vs. r with current q₁ and q₂ values, mark current position

Multiple Charges and Vector Addition

When multiple charges are present, the total electric force on any charge is the vector sum of all individual forces. This principle, called the principle of superposition, allows us to analyze complex charge configurations.

For example, if charge q~1~ experiences forces F~12~ (from q~2~) and F~13~ (from q~3~), the total force is:

$$\vec{F}{total} = \vec{F}$$} + \vec{F}_{13

Remember that forces are vectors, so you must account for both magnitude and direction. Forces pointing in opposite directions partially cancel, while forces in the same direction add together.

The Electric Field

The concept of electric force between charges is powerful, but it raises a puzzling question: How does one charge "know" about another charge that's some distance away? The answer lies in one of physics' most elegant ideas: the electric field.

What Is an Electric Field?

An electric field is a region of space around a charged object where other charged objects experience a force. Rather than thinking of charges exerting forces directly on each other across empty space, we say that a charge creates an electric field around itself, and this field then exerts a force on other charges within it.

Think of it like a magnetic field around a magnet, or the gravitational field around Earth. The field exists whether or not there's another charge present to feel it. It's a property of space itself, modified by the presence of charge.

Mathematically, the electric field E at any point is defined as the force per unit charge that would be experienced by a small positive test charge placed at that point:

$$\vec{E} = \frac{\vec{F}}{q_0}$$

where q~0~ is the magnitude of the test charge and F is the force it experiences.

For a single point charge Q, the electric field at distance r is:

$$E = k \frac{|Q|}{r^2}$$

The direction of the field is:

• Away from positive charges (positive charges "push" positive test charges away)
• Toward negative charges (negative charges "pull" positive test charges toward them)

Electric Field Lines

Electric field lines are a visual tool for representing electric fields. These imaginary lines show the direction and relative strength of the field at different locations.

Rules for drawing field lines:

1. Field lines point in the direction a positive test charge would move
2. Field lines start on positive charges and end on negative charges
3. The density of lines indicates field strength (more lines = stronger field)
4. Field lines never cross each other
5. Field lines are perpendicular to conductor surfaces

Common field line patterns:

• Single positive charge: Lines radiate outward in all directions
• Single negative charge: Lines converge inward from all directions
• Two equal opposite charges (dipole): Lines curve from positive to negative
• Two equal positive charges: Lines repel each other and curve away
• Parallel plates: Uniform field (straight parallel lines) between plates

Diagram: Electric Field Lines Visualization MicroSim

<summary>Electric Field Lines Visualization MicroSim</summary>
Type: microsim

Learning objective: Demonstrate electric field line patterns for various charge configurations

Canvas layout (900x700px):
- Main visualization (900x600): Shows charges and field lines
- Configuration selector (900x100): Choose different charge arrangements

Visual elements:
- Charged particles shown as colored circles (red = positive, blue = negative)
- Electric field lines drawn as smooth curves with arrowheads
- Field strength indicated by line density and color gradient (red = strong, yellow = medium, green = weak)
- Optional: Background color showing field magnitude

Charge configurations (buttons to select):
1. Single positive charge
2. Single negative charge
3. Electric dipole (equal ± charges)
4. Two positive charges
5. Two negative charges
6. Three charges in triangle
7. Parallel plates (uniform field)
8. Custom (user places charges)

Interactive controls:
- Configuration buttons
- Slider: Number of field lines (8 to 32)
- Checkbox: "Show field magnitude background"
- Checkbox: "Show equipotential lines"
- Button: "Clear all charges" (for custom mode)
- Click to place charges in custom mode (shift+click for negative)

Default parameters:
- Configuration: Electric dipole
- Field lines: 16
- Show magnitude: OFF
- Show equipotential: OFF

Behavior:
- Field lines calculated numerically from charge positions
- Lines start from positive charges, end on negative charges (or canvas edge)
- Smooth curves drawn using spline interpolation
- Color gradient along lines based on local field strength
- Equipotential lines (if enabled) shown perpendicular to field lines

Additional features:
- Drag charges to reposition (field lines update in real-time)
- Hover over any point to see field vector and magnitude
- Animation option: Show test charge moving along field line

Implementation notes:
- Use p5.js for rendering
- Calculate field at each point: E = Σ k·q/r² (vector sum)
- Use Runge-Kutta or Euler method to trace field lines
- Color gradient: map log(|E|) to color scale

Field Strength

Field strength (also called electric field magnitude) measures how strong the electric field is at a given location. It's measured in newtons per coulomb (N/C) or, equivalently, volts per meter (V/m).

The field strength determines how much force a charge will experience:

$$F = qE$$

A 1-coulomb charge in a field of strength 10 N/C experiences a 10-newton force. A 2-coulomb charge in the same field experiences a 20-newton force.

Field strength decreases with distance from a point charge following the inverse square law:

• At distance r: E = k Q/r^2^
• At distance 2r: E = k Q/(2r)^2^ = E~0~/4
• At distance 3r: E = k Q/(3r)^2^ = E~0~/9

Near the surface of everyday charged objects, field strengths typically range from 10^3^ to 10^6^ N/C. However, air breaks down (ionizes) at about 3 × 10^6^ N/C, creating a conducting path—this is what happens in lightning bolts and sparks.

Location	Typical Field Strength (N/C)
Near rubbed balloon	10^3^ - 10^4^
Inside typical thundercloud	10^4^ - 10^5^
Air breakdown threshold	3 × 10^6^
Inside atoms	10^11^

Electric Potential Energy

Electric forces do work when they move charges, just as gravitational forces do work when they lift or lower objects. This work is associated with a form of stored energy called electric potential energy.

Understanding Electric Potential Energy

Electric potential energy is the energy stored in a system of charged objects due to their positions relative to each other. When charges are in configurations that the electric force would change (like charges being pulled together or like charges being pushed apart), the system has electric potential energy.

The concept is directly analogous to gravitational potential energy:

• Lifting a mass against gravity increases gravitational PE
• Pushing like charges together against electric repulsion increases electric PE
• Pulling opposite charges apart against electric attraction increases electric PE

For two point charges q~1~ and q~2~ separated by distance r, the electric potential energy is:

$$U = k \frac{q_1 q_2}{r}$$

Key features of this equation:

• Positive energy (U > 0): Like charges (both positive or both negative) have positive PE, which increases as they get closer
• Negative energy (U < 0): Opposite charges have negative PE, which becomes more negative as they get closer
• Zero energy: U = 0 when charges are infinitely far apart (our reference point)
• 1/r dependence: Unlike force (1/r^2^), energy depends on 1/r

Energy in Electric Fields

When a charge moves through an electric field, the field does work on the charge, changing its kinetic energy. The work done equals the negative change in electric potential energy:

$$W = -\Delta U = -(U_f - U_i)$$

If the field does positive work (W > 0), the charge's kinetic energy increases and its potential energy decreases. This is exactly what happens when you release a ball and let gravity accelerate it downward—gravitational PE converts to kinetic energy.

For a charge q moving through a region where the electric field does work W, the change in kinetic energy is:

$$\Delta KE = W = -\Delta U$$

This is just conservation of energy for electric systems.

Diagram: Electric Potential Energy Interactive Chart

<summary>Electric Potential Energy Interactive Chart</summary>
Type: chart

Chart type: Line graph with multiple curves

Purpose: Show how electric potential energy varies with distance for different charge combinations

X-axis: Distance r (meters, from 0.1 to 5.0 m)
Y-axis: Electric potential energy U (joules, range -10 to +10 J)

Data series:
1. Two positive charges (red curve): q₁ = +2 μC, q₂ = +2 μC
   - U is positive (repulsive)
   - U decreases as r increases (1/r curve)
   - Asymptotically approaches zero

2. Two negative charges (blue curve): q₁ = -2 μC, q₂ = -2 μC
   - U is positive (repulsive)
   - Same shape as red curve

3. Opposite charges (purple curve): q₁ = +2 μC, q₂ = -2 μC
   - U is negative (attractive)
   - U increases (becomes less negative) as r increases
   - Asymptotically approaches zero from below

Title: "Electric Potential Energy vs. Distance"

Legend: Position top-right with charge combinations

Interactive features:
- Hover over curves to see exact U value at each distance
- Sliders to adjust charge magnitudes (updates curves in real-time)
- Vertical line showing current r value (draggable)
- Display box showing U values for all three cases at current r
- Toggle: "Show 1/r reference line"

Annotations:
- Horizontal dashed line at U = 0 labeled "Zero PE reference"
- Arrow indicating "Like charges have positive PE"
- Arrow indicating "Opposite charges have negative PE"
- Note: "PE → 0 as r → ∞"

Implementation: Chart.js with custom interactivity using JavaScript

Electric Potential and Voltage

While electric potential energy tells us about the energy stored in a system of charges, electric potential (often called voltage) is a property of a location in space.

Electric Potential

Electric potential at a point in space is the electric potential energy per unit charge at that location. It's defined as:

$$V = \frac{U}{q}$$

where V is electric potential (in volts, V), U is potential energy (in joules, J), and q is the charge (in coulombs, C).

For a single point charge Q, the electric potential at distance r is:

$$V = k \frac{Q}{r}$$

Note that potential is a scalar quantity (no direction), not a vector. This makes calculations much simpler than working with electric field vectors.

Key points about electric potential:

• Potential is highest near positive charges and lowest near negative charges
• Charges naturally move from high potential to low potential (like water flowing downhill)
• The potential difference between two points determines how much work is done moving a charge between them
• We often choose V = 0 at infinity (or sometimes at ground/Earth)

Voltage

Voltage (or potential difference) is the difference in electric potential between two points:

$$\Delta V = V_B - V_A$$

When we say a battery provides 9 volts, we mean there's a 9-volt potential difference between its terminals. If you move 1 coulomb of charge through this potential difference, 9 joules of energy is transferred:

$$W = q \Delta V$$

This is why voltage is measured in volts (V), where 1 volt = 1 joule per coulomb.

Common voltage examples:

• AA battery: 1.5 V
• Car battery: 12 V
• Household outlet (US): 120 V
• High-voltage power lines: 100,000 - 750,000 V
• Lightning bolt: 100 million V
• Static shock from carpet: 5,000 - 25,000 V

Diagram: Voltage and Electric Field Relationship Diagram

<summary>Voltage and Electric Field Relationship Diagram</summary>
Type: diagram

Purpose: Show the relationship between electric field, potential difference, and work done on charges

Components to show:
- Two parallel charged plates (top plate positive, bottom plate negative)
- Uniform electric field lines between plates (pointing downward)
- Positive test charge at three positions: near top plate, middle, near bottom plate
- Potential values labeled at different heights (V_high at top, V_low at bottom)
- Energy bar charts showing PE and KE of test charge at each position

Connections:
- Electric field vector E pointing downward
- Equipotential lines (dashed horizontal lines) labeled with voltages
- Arrow showing direction of decreasing potential
- Path of test charge moving downward (with kinetic energy arrow increasing)

Style: Cross-sectional view with clear labeling

Labels:
- "Positive plate (+)" at top
- "Negative plate (-)" at bottom
- "Electric field E (uniform)"
- "High potential V_high"
- "Low potential V_low"
- "ΔV = V_high - V_low"
- Energy bars: "PE" (decreases going down), "KE" (increases going down)
- "Total energy constant"

Mathematical relationships shown:
- E = ΔV / d (where d is plate separation)
- W = qΔV (work done on charge)
- ΔPE = -qΔV (change in potential energy)

Color scheme: Positive plate in red, negative plate in blue, field lines in gold, equipotential lines in green

The Relationship Between Field and Potential

Electric field and potential are intimately related. In fact, the electric field is the negative gradient (rate of change) of the potential:

$$E = -\frac{\Delta V}{\Delta d}$$

For a uniform field (like between parallel plates), this simplifies to:

$$E = \frac{V}{d}$$

where V is the potential difference and d is the distance.

This relationship reveals a profound connection:

• Strong fields exist where potential changes rapidly with distance (steep potential "hill")
• Weak fields exist where potential changes gradually (gentle potential slope)
• Zero field exists where potential is constant (flat potential "plateau")

For example, between parallel plates separated by 10 cm with a potential difference of 100 V:

$$E = \frac{100 \text{ V}}{0.1 \text{ m}} = 1000 \text{ V/m} = 1000 \text{ N/C}$$

This relationship makes problem-solving easier: sometimes it's simpler to calculate potential (a scalar) than electric field (a vector), then use the relationship to find the field.

Connections to Prior Knowledge

The concepts in this chapter build on and parallel ideas you've encountered earlier in the course.

Force Laws and Fields

Coulomb's Law for electric force has the same mathematical form as Newton's law of gravitation:

Property	Gravitational Force	Electric Force
Formula	F = G m₁m₂/r²	F = k q₁q₂/r²
Nature	Always attractive	Attractive or repulsive
Relative strength	Extremely weak	Very strong
Constant	G = 6.67 × 10^-11^	k = 8.99 × 10^9^
Units	N	N

Both forces:

• Follow inverse square laws (1/r^2^)
• Act at a distance
• Can be described using field concepts
• Have associated potential energy

Energy Conservation

The work-energy theorem applies to electric forces just as it does to mechanical forces:

$$W_{net} = \Delta KE$$

For conservative forces (including electric forces between static charges):

$$\Delta KE + \Delta PE = 0$$

This means that as a charge accelerates in an electric field, it trades potential energy for kinetic energy, just as a falling object trades gravitational PE for KE.

Vectors and Fields

The electric field is a vector field—it has both magnitude and direction at every point in space. Working with electric fields requires the same vector addition skills you developed earlier:

• Breaking vectors into components
• Adding vectors head-to-tail
• Using the Pythagorean theorem to find resultant magnitudes
• Using trigonometry to find directions

Real-World Applications

Understanding electric charge and fields is essential for countless technologies.

Lightning and Lightning Protection

Lightning is one of nature's most dramatic demonstrations of electric charge in action. Inside thunderclouds, rising and falling air currents cause ice particles and water droplets to collide, transferring electrons and separating charge. The bottom of the cloud becomes negatively charged, while the top becomes positive.

This charge separation creates an enormous electric field—up to millions of volts per meter. When the field becomes strong enough to ionize air (about 3 × 10^6^ N/C), a conducting channel forms, allowing electrons to rush from cloud to ground (or ground to cloud, or cloud to cloud) in a fraction of a second.

Lightning rods protect buildings by providing a preferred, low-resistance path to ground. The rod is typically copper or aluminum, mounted at the highest point of a structure, and connected via thick cables to grounding rods buried deep in the earth. When lightning strikes, the enormous current (typically 20,000 to 200,000 amperes) flows through the conductor system rather than through the building, preventing fire and structural damage.

Electrostatic Applications

Many everyday technologies exploit electrostatic principles:

Photocopiers and laser printers use electric charge to transfer toner to paper:

1. A drum is given a uniform positive charge
2. A laser removes charge from specific areas, creating an image pattern
3. Negatively charged toner particles stick to the remaining positive areas
4. The toner is transferred to paper and fused with heat

Air purifiers use electric fields to remove particles:

1. Airborne particles pass through a charging section with a high-voltage wire
2. Particles gain electrons and become negatively charged
3. Charged particles are attracted to positive collection plates
4. Accumulated particles are washed or wiped away

Paint spraying in automotive factories uses electrostatic charging:

1. Paint droplets are given an electric charge
2. The metal car body is oppositely charged (or grounded)
3. Electric attraction pulls paint droplets toward the car
4. Result: more uniform coating with less waste

Van de Graaff generators demonstrate charge accumulation:

1. A moving belt transfers charge to a hollow metal sphere
2. Charge accumulates on the outside of the sphere
3. Extremely high voltages (millions of volts) can be reached
4. Used for physics demonstrations and in particle accelerators

Electronic Devices

Every electronic device—smartphones, computers, televisions, medical equipment—relies on controlling the movement of charge. Semiconductors allow engineers to create components where tiny voltages control large currents, enabling:

• Transistors: Switches that can turn on and off billions of times per second
• Diodes: One-way valves for electric current
• Solar cells: Converting light energy to electric energy
• LEDs: Converting electric energy to light with high efficiency

The principles of electric fields, potential, and energy underlie all circuit behavior, energy storage (in capacitors), and signal processing.

Summary and Key Takeaways

Electric charge is one of the fundamental properties of matter, existing in positive and negative varieties that can attract or repel each other. The conservation of charge—one of physics' most fundamental laws—ensures that charge can be transferred but never created or destroyed.

Materials are classified by their response to electric forces:

• Conductors allow charge to flow freely
• Insulators resist charge flow
• Semiconductors have controllable conductivity
• Superconductors conduct with zero resistance below a critical temperature

Objects can be charged through three mechanisms:

• Friction (rubbing materials together transfers electrons)
• Contact (touching a charged object shares charge)
• Induction (using a nearby charge to redistribute charge, then isolating part of the object)

Coulomb's Law quantifies the electric force between charges, revealing an inverse square relationship similar to gravity. Electric fields provide an elegant way to think about these forces—charges create fields that extend through space, and these fields exert forces on other charges.

Electric potential energy represents the energy stored in configurations of charges. Electric potential (voltage) is the potential energy per unit charge, and potential differences drive the flow of charge in circuits. The electric field and potential are intimately connected: strong fields exist where potential changes rapidly with position.

These concepts form the foundation for understanding electricity and all electrical technology. In future physics studies, you'll explore how moving charges create magnetic fields (electromagnetism), how changing magnetic fields generate electric fields (electromagnetic induction), and how electric and magnetic fields combine to create electromagnetic waves—including visible light.

Practice Problems

1. Two charges of +3.0 μC and -5.0 μC are separated by 0.40 m. Calculate the magnitude of the electric force between them.

2. A plastic rod rubbed with wool acquires a charge of -2.0 × 10^-7^ C. How many electrons were transferred to the rod?

3. The electric field between two parallel plates separated by 5.0 cm is 4000 N/C. What is the potential difference between the plates?

4. An electron (charge -1.6 × 10^-19^ C, mass 9.1 × 10^-31^ kg) is accelerated from rest through a potential difference of 100 V. What is its final kinetic energy? What is its final speed?

5. Three charges are arranged at the corners of an equilateral triangle with sides of 0.30 m. The charges are: q~1~ = +4.0 μC, q~2~ = +4.0 μC, and q~3~ = -4.0 μC. Calculate the net electric force on q~3~.

Further Reading

• Feynman, Richard P. (2011). The Feynman Lectures on Physics, Vol. II: Mainly Electromagnetism and Matter. Basic Books.
• Griffiths, David J. (2017). Introduction to Electrodynamics (4th ed.). Cambridge University Press.
• Purcell, Edward M., & Morin, David J. (2013). Electricity and Magnetism (3rd ed.). Cambridge University Press.
• Walker, Jearl, Halliday, David, & Resnick, Robert (2014). Fundamentals of Physics (10th ed.). Wiley.

Looking Ahead

In the next chapter, we'll explore what happens when charges move continuously through materials—electric current. You'll learn about:

• Current and current density
• Resistance and Ohm's Law
• Series and parallel circuits
• Kirchhoff's circuit laws
• Power dissipation in circuits
• Real-world circuit analysis

The concepts of electric potential and voltage you've learned here will become essential tools for understanding how energy flows through electrical circuits and powers the devices we use every day.