Electrostatics and Circuits Unit
An NGSS-aligned unit covering electrostatics, Coulomb's Law, electric fields, voltage, capacitance, Ohm's Law, series & parallel circuits, Kirchhoff's Laws, and circuit inquiry. Each section is tagged with the NGSS Performance Expectation(s), Science & Engineering Practice(s), Disciplinary Core Idea(s), and Crosscutting Concept(s) it addresses.
NGSS Performance Expectations
This unit primarily addresses high school NGSS standards in HS-PS2: Motion and Stability: Forces and Interactions and HS-PS3: Energy. Every section below maps to at least one Performance Expectation, Science & Engineering Practice, Disciplinary Core Idea, and Crosscutting Concept.
HS-PS2-4 Coulomb's Law
- Use mathematical representations of Newton's Law of Gravitation and Coulomb's Law to describe and predict the gravitational and electrostatic forces between objects.
HS-PS3-5 Fields and Energy
- Develop and use a model of two objects interacting through electric or magnetic fields to illustrate the forces between objects and the changes in energy of the objects due to the interaction.
HS-PS2-5 Electromagnetism
- Plan and conduct an investigation to provide evidence that an electric current can produce a magnetic field and that a changing magnetic field can produce an electric current.
HS-PS3 Energy Connections
- Use conservation of energy, energy transfer, and mathematical modeling to analyze electrical systems, power, and storage.
Science & Engineering Practices
- SEP Using Mathematics and Computational Thinking
- SEP Developing and Using Models
- SEP Planning and Carrying Out Investigations
- SEP Analyzing and Interpreting Data
- SEP Constructing Explanations and Designing Solutions
Disciplinary Core Ideas
- PS2.B Types of Interactions
- PS3.A Definitions of Energy
- PS3.B Conservation of Energy and Energy Transfer
- PS3.C Relationship Between Energy and Forces
Crosscutting Concepts
- CCC Cause and Effect
- CCC Systems and System Models
- CCC Energy and Matter
- CCC Patterns
HS-PS2-4 HS-PS3-5 Developing Models PS2.B / PS3.C Cause & Effect
Electrostatics and the American Experiment
Benjamin Franklin, lightning, charge, and the revolutionary power of scientific thinking
1. Historical Introduction to Electrostatics
Physics, philosophy, and religion have often overlapped in meaningful ways. In the 1700s, lightning was still viewed by many people as mysterious, supernatural, or even divine. Benjamin Franklin’s work in electrostatics helped move the explanation of lightning from fear and mystery toward evidence, experiment, and physical law.
Franklin was not simply a scientist. He was a printer, writer, inventor, businessman, diplomat, and revolutionary. As the son of a candle and soap maker with limited formal schooling, Franklin became a symbol of the self-made person. His rise helped shape the image of the ordinary individual using knowledge, discipline, and practical intelligence to make a mark on the world.
His work in electricity gave him credibility in Europe. His work as a printer, propagandist, and diplomat helped fuel the American Revolution. Franklin’s life shows how science, communication, politics, and economics can become deeply connected.
2. What Is Electrostatics?
Electrostatics is the study of electric charge at rest. Before students analyze full circuits, current, resistance, and capacitors, they need to understand what charge is and why separated charge can create forces, energy differences, and motion.
Core Ideas
- Objects can become charged by gaining or losing electrons.
- Like charges repel and opposite charges attract.
- Charge can be transferred by contact.
- Charge can be redistributed without contact through induction.
- Electric potential difference, or voltage, describes energy per unit charge.
Physics Lens
Electrostatics is not only about sparks. It is about force, energy, and motion. A charged object creates an electric field. Another charge placed in that field experiences a force. If that charge is free to move, electric potential energy can become kinetic energy.
3. Franklin’s Model of Electricity
Franklin proposed that electricity behaved like a kind of invisible fluid. While modern physics explains electricity through particles, fields, and charge interactions, Franklin’s model was powerful because it gave people a way to reason about charge transfer and conservation.
Franklin’s Lasting Contributions
- He helped popularize the terms positive and negative charge.
- He argued that electrical effects could be investigated experimentally.
- He connected lightning to electrical discharge.
- He developed the lightning rod as a practical application of electrostatics.
4. Franklin, Lightning, and the Kite Experiment
Franklin’s famous kite experiment is one of the best-known stories in the history of science. The basic idea was to show that storm clouds carried electrical charge. A wet string, a key, and a conducting path allowed charge from the storm environment to be detected.
Why the Experiment Mattered
The kite experiment mattered because it connected a frightening natural event to a physical process that could be studied, predicted, and controlled. Lightning was no longer only a symbol of divine anger or mystery. It became evidence of charge separation and electrical discharge in the atmosphere.
5. Lightning Rods and Public Reaction
Franklin’s lightning rod was a practical invention based on electrostatics. A conducting rod attached to a building gives charge a safer path to the ground. Instead of allowing lightning to pass through wood, brick, or people, the rod directs the electrical discharge into Earth.
This invention created public debate. Some people welcomed it as a life-saving application of science. Others objected because they believed lightning was an act of God that humans should not interfere with. This disagreement reveals an important historical shift: scientific explanation was beginning to challenge older forms of authority.
6. Electrostatics and Revolution
Franklin’s scientific reputation made him famous in Europe before the American Revolution. When he later served as a diplomat in France, his image as a philosopher, scientist, and practical genius helped him gain attention and respect.
Franklin also understood the power of communication. Through printing, writing, satire, and public argument, he helped shape colonial identity. His scientific work and political work both depended on the same habit of mind: question inherited explanations, gather evidence, communicate clearly, and act.
Big Connection
Franklin’s work asks a larger question: if nature can be understood through reason instead of fear, can government and society also be redesigned through reason instead of inherited authority?
7. Inquiry Assignment — Stealing God’s Thunder
In this inquiry, you will investigate how Franklin’s work with electricity influenced science, religion, public opinion, and the American Revolution. Your goal is not to write a simple biography. Your goal is to explain how scientific ideas can change culture.
Research Questions
Franklin and Science
- What did scientists believe about electricity before Franklin?
- What did Franklin contribute to the study of electrostatics?
- Why was the kite experiment important?
- How does a lightning rod work?
- Why do the terms positive and negative charge matter?
Franklin and Society
- Why did some people object to lightning rods?
- How did Franklin represent Enlightenment thinking?
- How did Franklin’s personal background shape his public image?
- How did his scientific fame help him politically?
- Why does scientific literacy matter in a democracy?
8. Electrostatics Diagram Tasks
Create labeled diagrams and written explanations for each process below. Each explanation should connect the diagram to force, energy, or charge movement.
A. Charging by Friction
Show how rubbing two materials together can transfer electrons from one object to another.
B. Charging by Contact
Show how a charged object can transfer charge to a neutral object through direct contact.
C. Charging by Induction
Show how a charged object can cause charge separation in another object without touching it.
D. Grounding
Show how Earth can act as a large charge reservoir, allowing excess charge to move into or out of an object.
9. Historical Reflection Essay
Write a 2–3 page response to the following prompt:
Your response should include:
- At least three historical sources
- An explanation of Franklin’s work in electrostatics
- A discussion of Enlightenment thinking
- A connection to the American Revolution
- Your own argument about why Franklin’s work mattered
Optional Writing Structure
- Introduction: Introduce Franklin and your main claim.
- Science Paragraph: Explain his work with electricity and lightning.
- Society Paragraph: Explain public reaction and religious or cultural conflict.
- Revolution Paragraph: Connect Franklin’s reputation to politics and diplomacy.
- Conclusion: Explain what Franklin’s story reveals about science and society.
10. Suggested Sources
- The Franklin Institute — Franklin’s Kite and Key Experiment
- The Franklin Institute — Franklin’s Lightning Rod
- Library of Congress — Benjamin Franklin: Business and Science
- Library of Congress — Benjamin Franklin Papers
11. Assessment Rubric
| Category | Developing | Proficient | Advanced |
|---|---|---|---|
| Historical Research | Uses limited or weak evidence. | Uses relevant sources to explain Franklin’s work. | Uses strong evidence to connect Franklin to science, society, and revolution. |
| Physics Understanding | Explains charge or electrostatics with major gaps. | Correctly explains major electrostatic concepts. | Clearly connects charge, force, energy, fields, and grounding. |
| Diagrams | Diagrams are missing or unclear. | Diagrams are accurate and labeled. | Diagrams are detailed, labeled, and connected to written explanations. |
| Analysis | Mostly summarizes Franklin’s life. | Explains why Franklin’s work mattered. | Develops a strong argument about science, authority, and revolution. |
| Writing Quality | Writing is difficult to follow. | Writing is organized and clear. | Writing is polished, persuasive, and thoughtful. |
12. Closing Thought
Franklin’s work with electricity helped people see that nature could be investigated rather than feared. That shift mattered far beyond physics. It helped support a larger Enlightenment belief that human beings could use reason to improve society.
The question remains important today: when scientific understanding changes, how should society change with it?
HS-PS2-4 HS-PS3-5 HS-PS2-5 Planning Investigations PS2.B Cause & Effect
Electrostatics Demonstrations Companion Guide
Charge, induction, electric fields, and the foundations of electricity through classroom demonstrations.
1. Core Vocabulary and Concepts
These demonstrations are designed to make invisible electric interactions visible. As you watch each demonstration, focus on where charge is moving, where charge is separating, and what evidence shows that an electric field is present.
Charge
A property of matter responsible for electrical forces. Charges may be positive or negative.
Electron
A negatively charged particle. In most everyday charging situations, electrons are the particles that move.
Induction
The redistribution of charge caused by a nearby charged object without direct contact.
Grounding
Connecting an object to Earth so electrons can move into or out of the object.
Conductor
A material in which charge can move easily, such as metal.
Insulator
A material in which charge does not move easily, such as rubber, plastic, or dry glass.
Electric Field
A region around a charged object where electric forces act on other charges.
Potential Difference
A difference in electrical energy per unit charge. This is commonly called voltage.
2. Demonstration: Charging by Induction
In charging by induction, a charged object causes electrons in another object to move without touching it directly. This shows that electric forces can act through space.
A nearby charged object redistributes charge in a conductor without touching it.
What to Watch For
- Does the object become attracted before contact?
- Does charge separation happen without touching?
- What changes if the object is grounded?
Conceptual Questions
Why does charge move even though the rod never touches the sphere?
The electric field from the charged rod exerts forces on electrons inside the conductor, causing them to redistribute.
Why is grounding important in induction charging?
Grounding allows electrons to leave or enter the object. This can leave the object with a net charge after the charged rod is removed.
Would induction work as effectively with an insulator?
No. Charges in insulators cannot move freely through the material, so large-scale redistribution is limited.
3. Demonstration: Van de Graaff Generator
A Van de Graaff generator uses a moving belt to transport charge onto a metal dome. Because the dome is a conductor, excess charge spreads over its outer surface.
A moving belt carries charge to the dome, where it spreads over the conductor’s surface.
What to Watch For
- Hair standing up or lightweight objects being repelled.
- Sparks forming when the electric field becomes strong enough.
- Charge spreading across the dome instead of staying in one location.
Conceptual Questions
Why does charge move to the outer surface of the dome?
Charges repel one another and spread as far apart as possible on a conductor.
Why can sparks jump from the dome?
The electric field can become strong enough to ionize air, allowing charge to rapidly move through the air gap.
Why is dry air better for Van de Graaff demonstrations?
Moisture in humid air allows charge to leak away more easily, reducing the buildup of static charge.
4. Demonstration: Franklin’s Bells
Franklin’s bells demonstrate charge transfer and repeated motion caused by electrostatic attraction and repulsion. A small conducting ball moves between two oppositely charged bells.
The ball alternates between attraction, contact, charge transfer, and repulsion.
What to Watch For
- The ball is first attracted to one bell.
- After touching, it takes on the same type of charge as that bell.
- It is then repelled and attracted to the opposite bell.
Conceptual Questions
Why does the ball begin moving?
The neutral ball becomes polarized near a charged bell and experiences attraction.
Why does the ball reverse direction after touching the bell?
After contact, the ball gains the same type of charge as the bell and is repelled.
Why does the process continue repeatedly?
The ball alternates charge through repeated contact with oppositely charged bells.
5. Demonstration: Faraday’s Ice Pail
Faraday’s ice pail experiment shows how charge behaves on conductors. When a charged object is lowered into a metal container, charge redistributes through the conducting surface.
A conductor redistributes charge so that excess charge resides on the outside surface.
What to Watch For
- The pail responds even before the charged object touches it.
- Charge appears on the outside of the conductor.
- The result connects directly to electrostatic shielding and Faraday cages.
Conceptual Questions
Why does the outer surface become charged?
Charges in a conductor repel each other and move until electrostatic equilibrium is reached.
How does this relate to a Faraday cage?
External electric fields cause charges to redistribute on the outside surface, shielding the interior region.
Why are passengers in a car relatively safe during lightning strikes?
The conducting shell of the car redirects charge around the outside of the vehicle, reducing the electric field inside.
6. Demonstration: Lenz’s Law
Lenz’s Law describes how induced currents oppose changes in magnetic flux. Although this demonstration belongs to electromagnetism rather than pure electrostatics, it shows an important extension: moving charge can create magnetic effects, and changing magnetic fields can cause charge to move.
Changing magnetic flux induces currents that oppose the motion of the magnet.
What to Watch For
- The magnet falls more slowly through a conducting tube than through air.
- The tube may not be magnetic, but currents are induced in it.
- The motion demonstrates conservation of energy.
Conceptual Questions
Why does the magnet fall slowly through the tube?
The changing magnetic field induces currents in the conductor, and those currents create magnetic fields opposing the motion.
Where does the lost kinetic energy go?
The energy is converted into thermal energy due to electrical resistance in the conductor.
Why is this related to conservation of energy?
If induced currents helped the motion instead of opposing it, energy would appear from nowhere.
7. Reflection Questions
How are all of these demonstrations connected?
Each demonstration involves electric charge responding to forces, fields, or changing energy conditions.
What evidence suggests that electric fields are physically real?
Electric fields produce measurable forces and motion without direct contact between objects.
Why did demonstrations like these help convince people that electricity followed physical laws?
The effects were repeatable, measurable, and predictable through experiment and mathematical reasoning.
How does this connect back to Franklin?
Franklin’s work helped show that electrical phenomena such as sparks and lightning could be explained by physical principles rather than treated only as mysterious or supernatural events.
HS-PS2-4 HS-PS3-5 HS-PS3 Using Math PS2.B / PS3.B / PS3.C Systems & Models
Foundations of Electricity and Circuits
A physics-centered primer connecting charge, force, fields, voltage, current, capacitance, resistance, and circuit analysis.
What You Should Be Able To Do
- Explain electric charge as a conserved property of matter.
- Use Coulomb’s Law to calculate the force between point charges.
- Interpret electric fields as force per unit positive test charge.
- Distinguish electric potential energy from electric potential, or voltage.
- Use capacitance to describe charge storage and energy storage.
- Use current, voltage, and resistance correctly in Ohm’s Law.
- Analyze series, parallel, and series–parallel resistor circuits.
- Compare equivalent resistance methods with Kirchhoff’s loop and junction laws.
- Recognize when a circuit can be simplified and when Kirchhoff’s Laws are the better tool.
Glossary of Electricity Terms
Charge, \(q\)
A fundamental property of matter. Charge is measured in coulombs. Protons are positive, electrons are negative.
Conservation of Charge
Total charge in a closed system remains constant. Charge can move, but it is not created or destroyed in ordinary circuit processes.
Electric Force, \(F_e\)
The attractive or repulsive force between charged objects.
Electric Field, \(E\)
Force per unit positive test charge. Electric fields point in the direction a positive test charge would accelerate.
Electric Potential Energy, \(U_e\)
Energy stored because of the position of charges in an electric field.
Voltage, \(V\)
Electric potential difference. It measures energy transferred per coulomb of charge.
Current, \(I\)
The rate of charge flow. Current is measured in amperes, where \(1\text{ A}=1\text{ C/s}\).
Resistance, \(R\)
A measure of how strongly a component opposes current.
Capacitance, \(C\)
A measure of how much charge a device stores per volt of potential difference.
Node or Junction
A point where current can split or combine.
Loop
A closed path through a circuit. Kirchhoff’s loop rule applies around closed paths.
Equivalent Resistance
A single resistance that would draw the same current from the same source as the original resistor network.
1. Electric Charge
HS-PS2-4 Using Math PS2.B Patterns
Electric charge is the starting point for all electrical phenomena. Matter contains positive protons and negative electrons. In most everyday charging processes, electrons are the particles that move from one object to another.
A neutral object has equal amounts of positive and negative charge. A negatively charged object has excess electrons. A positively charged object has lost electrons.
\(q\) = total charge, \(n\) = number of elementary charges, and \(e = 1.60\times10^{-19}\ \text{C}\).
Charge interaction is the foundation for electric force, field, potential, and current.
Check Understanding: Charge
Conceptual Question: A balloon rubbed on hair sticks to a neutral wall. How can a charged balloon attract a neutral wall?
The charged balloon polarizes molecules in the wall. Charges within the wall shift slightly so the side closer to the balloon has the opposite effective charge. The wall remains neutral overall, but the closer opposite charge creates a stronger attraction than the farther like charge creates repulsion.
Calculation Question: How many excess electrons are on an object with charge \(-3.20\times10^{-9}\ \text{C}\)?
Use \(q=ne\). The number of electrons is \[ n=\frac{|q|}{e}=\frac{3.20\times10^{-9}}{1.60\times10^{-19}}=2.00\times10^{10} \] The object has \(2.00\times10^{10}\) excess electrons.
2. Electric Force and Electric Fields
HS-PS2-4 HS-PS3-5 Using Math / Developing Models PS2.B / PS3.C Cause & Effect
Charged objects exert forces on one another. Coulomb’s Law gives the size of the force between two point charges. The force increases with the size of each charge and decreases with the square of the distance between them.
\(k = 8.99\times10^9\ \text{N}\cdot\text{m}^2/\text{C}^2\)
The electric field shifts our attention from “one charge pushing another charge” to “a charged object changing the space around it.” If a test charge is placed in that space, it experiences a force.
The first equation defines electric field. The second gives the field caused by a point charge \(Q\).
Electric field lines point away from positive charges and toward negative charges.
Check Understanding: Force and Fields
Conceptual Question: Why do electric field lines never cross?
At any point in space, the electric field has one direction. If two field lines crossed, that point would have two different field directions, which is physically impossible.
Calculation Question: Two charges, \(+2.0\ \mu\text{C}\) and \(-3.0\ \mu\text{C}\), are separated by \(0.20\ \text{m}\). Find the force magnitude.
\[ F_e=k\frac{|q_1q_2|}{r^2} =(8.99\times10^9)\frac{(2.0\times10^{-6})(3.0\times10^{-6})}{(0.20)^2} \] \[ F_e\approx 1.35\ \text{N} \] Since the charges are opposite, the force is attractive.
Calculation Question: A \(4.0\times10^{-6}\ \text{C}\) test charge feels a \(0.80\ \text{N}\) force. What is the electric field?
\[ E=\frac{F}{q}=\frac{0.80}{4.0\times10^{-6}}=2.0\times10^5\ \text{N/C} \]
3. Electric Potential Energy, Electric Potential, and Voltage
HS-PS3-5 HS-PS3 Developing Models PS3.C Energy & Matter
Electric force describes pushes and pulls. Electric potential describes energy. A charge in an electric field can have electric potential energy, just as a mass held above the ground has gravitational potential energy.
Voltage is not the same thing as energy. Voltage is energy per unit charge. A 9 V battery gives \(9\ \text{J}\) of energy to each coulomb of charge that moves through it.
Voltage is energy per unit charge. One volt equals one joule per coulomb.
Gravitational Analogy
A ball rolls downhill because gravity pulls it from high gravitational potential energy to low gravitational potential energy.
Electrical Analogy
Positive charge naturally moves from high electric potential to low electric potential if it is free to move.
Check Understanding: Potential and Voltage
Conceptual Question: Why is voltage often compared to height?
Height helps describe gravitational potential energy per unit mass. Voltage describes electric potential energy per unit charge. In both cases, differences matter: objects move or energy transfers when there is a difference in potential.
Calculation Question: A battery transfers \(45\ \text{J}\) of energy to \(5.0\ \text{C}\) of charge. What is the voltage?
\[ V=\frac{W}{q}=\frac{45}{5.0}=9.0\ \text{V} \]
4. Capacitance and Stored Electrical Energy
HS-PS3-5 Developing Models PS3.C Systems & Models
A capacitor stores charge by separating positive and negative charge on two conductors. The simplest model is two parallel plates separated by an insulating material or air gap.
The battery does work to separate charge. That energy is stored in the electric field between the plates.
The final equation applies to an ideal parallel-plate capacitor.
A capacitor stores energy in the electric field between separated charges.
Check Understanding: Capacitance
Conceptual Question: Why does decreasing plate separation increase capacitance?
Bringing plates closer makes it easier for opposite charges on the two plates to attract and store more charge for the same voltage. In the parallel-plate model, \(C=\varepsilon_0A/d\), so decreasing \(d\) increases \(C\).
Calculation Question: A capacitor stores \(0.0040\ \text{C}\) at \(12\ \text{V}\). Find its capacitance.
\[ C=\frac{Q}{V}=\frac{0.0040}{12}=3.33\times10^{-4}\ \text{F} \]
Calculation Question: How much energy is stored in a \(2200\ \mu\text{F}\) capacitor charged to \(9.0\ \text{V}\)?
Convert \(2200\ \mu\text{F}=2.20\times10^{-3}\ \text{F}\). \[ U_C=\frac{1}{2}CV^2=\frac{1}{2}(2.20\times10^{-3})(9.0)^2 \] \[ U_C\approx 8.91\times10^{-2}\ \text{J} \]
5. Current, Resistance, and Ohm’s Law
HS-PS3 Using Math PS3.B Energy & Matter
Current is the rate of charge flow through a cross-section of a wire. Resistance describes how much a material or component opposes that flow.
Use \(\Delta V\) for potential difference across a circuit element.
A resistor does not destroy charge. Instead, it converts electrical energy into thermal energy, light, or another form. In a light bulb, electrical energy becomes thermal energy and visible light.
Check Understanding: Current and Resistance
Conceptual Question: Why do electrons drift slowly even though a light turns on almost immediately?
Individual electrons move slowly through a wire, but the electric field that pushes them is established through the circuit very quickly. The situation is similar to pushing one end of a long line of touching objects; motion appears at the far end even though each object moves only a little.
Calculation Question: \(18\ \text{C}\) of charge passes through a wire in \(6.0\ \text{s}\). What is the current?
\[ I=\frac{\Delta q}{\Delta t}=\frac{18}{6.0}=3.0\ \text{A} \]
Calculation Question: A \(12\ \text{V}\) battery drives \(0.50\ \text{A}\) through a resistor. Find the resistance and power.
\[ R=\frac{V}{I}=\frac{12}{0.50}=24\ \Omega \] \[ P=IV=(0.50)(12)=6.0\ \text{W} \]
6. Series Circuits
HS-PS3 Using Math PS3.B Systems & Models
In a series circuit, components are connected in a single path. Since there is only one path, the current is the same through each resistor. The voltage supplied by the battery is divided across the resistors.
Series circuits have one path for current.
Check Understanding: Series Circuits
Conceptual Question: Why does adding more resistors in series reduce the total current?
Adding resistors in series increases the equivalent resistance. For a fixed battery voltage, Ohm’s Law \(I=V/R\) says that a larger resistance produces a smaller current.
Calculation Question: A \(12\ \text{V}\) battery is connected to \(2\ \Omega\), \(4\ \Omega\), and \(6\ \Omega\) resistors in series. Find \(R_{\text{eq}}\), total current, and voltage drop across each resistor.
\[ R_{\text{eq}}=2+4+6=12\ \Omega \] \[ I=\frac{V}{R_{\text{eq}}}=\frac{12}{12}=1.0\ \text{A} \] Voltage drops: \[ V_1=(1.0)(2)=2\ \text{V},\quad V_2=(1.0)(4)=4\ \text{V},\quad V_3=(1.0)(6)=6\ \text{V} \] The voltage drops add to \(12\ \text{V}\).
7. Parallel Circuits
HS-PS3 Using Math PS3.B Systems & Models
In a parallel circuit, components are connected across the same two points. Each branch receives the same voltage, but the current splits among the branches.
Parallel circuits provide multiple paths for current.
Check Understanding: Parallel Circuits
Conceptual Question: Why does adding another branch in parallel usually decrease equivalent resistance?
A new branch gives current another path. More available paths make it easier for charge to flow from the battery, so the equivalent resistance decreases.
Calculation Question: A \(6\ \Omega\) resistor and a \(3\ \Omega\) resistor are connected in parallel to a \(12\ \text{V}\) battery. Find \(R_{\text{eq}}\), branch currents, and total current.
\[ \frac{1}{R_{\text{eq}}}=\frac{1}{6}+\frac{1}{3}=\frac{1}{6}+\frac{2}{6}=\frac{3}{6} \] \[ R_{\text{eq}}=2\ \Omega \] Each branch has \(12\ \text{V}\): \[ I_1=\frac{12}{6}=2\ \text{A},\quad I_2=\frac{12}{3}=4\ \text{A} \] \[ I_{\text{total}}=2+4=6\ \text{A} \]
8. Series–Parallel Circuits
HS-PS3 Using Math PS3.B Systems & Models
Many real circuits are combinations of series and parallel sections. The key is to simplify one piece at a time. Start with the most obvious series or parallel group, replace it with an equivalent resistance, and repeat until the circuit reduces to one total resistance.
Break a complex circuit into smaller series and parallel pieces.
Check Understanding: Series–Parallel Circuits
Conceptual Question: Why do real household circuits use parallel branches more than one long series path?
Parallel branches allow each device to receive the full household voltage and operate independently. If one device turns off or fails, the other branches can still operate.
Calculation Question: A \(4\ \Omega\) resistor is in series with two \(6\ \Omega\) resistors in parallel. The battery is \(12\ \text{V}\). Find total resistance and total current.
First reduce the parallel pair: \[ \frac{1}{R_p}=\frac{1}{6}+\frac{1}{6}=\frac{2}{6}=\frac{1}{3} \] \[ R_p=3\ \Omega \] Now add the series resistor: \[ R_{\text{eq}}=4+3=7\ \Omega \] Total current: \[ I_{\text{total}}=\frac{12}{7}=1.71\ \text{A} \]
Follow-Up: What voltage is across the parallel section?
The current through the \(4\ \Omega\) series resistor is \(1.71\ \text{A}\), so its voltage drop is \[ V_4=IR=(1.71)(4)=6.86\ \text{V} \] The remaining voltage is across the parallel section: \[ V_p=12-6.86=5.14\ \text{V} \]
9. Kirchhoff’s Laws
HS-PS3 Using Math PS3.B Energy & Matter
Equivalent resistance works beautifully for circuits that can be reduced into obvious series and parallel groups. But not all circuits simplify cleanly. Kirchhoff’s Laws are more general because they come directly from conservation laws.
Junction Rule
Current entering a junction equals current leaving the junction.
This is conservation of charge.
Loop Rule
Around any closed loop, voltage gains and voltage drops sum to zero.
This is conservation of energy.
Check Understanding: Kirchhoff’s Laws
Conceptual Question: Why is Kirchhoff’s loop rule a conservation of energy statement?
A charge that moves around a complete loop returns to its starting point. Its net change in electric potential energy must be zero. Therefore, voltage gains from batteries must balance voltage drops across circuit elements.
Calculation Question: Use Kirchhoff’s loop rule for a loop with a \(12\ \text{V}\) battery, a \(2\ \Omega\) resistor, and a \(4\ \Omega\) resistor in series.
Choose the direction of current around the loop: \[ +12-2I-4I=0 \] \[ 12-6I=0 \] \[ I=2.0\ \text{A} \] This matches the equivalent resistance method because \(R_{\text{eq}}=6\ \Omega\), so \(I=12/6=2.0\ \text{A}\).
10. Equivalent Resistance vs. Kirchhoff’s Loop Method
These two methods often produce the same result, but they represent different ways of thinking.
| Equivalent Resistance Method | Kirchhoff Method |
|---|---|
| Best when the circuit has clear series and parallel groups. | Best when the circuit has multiple loops, multiple batteries, or unclear simplification. |
| Replaces resistor groups with a single equivalent resistance. | Writes equations using current conservation and voltage conservation. |
| Usually faster for introductory resistor networks. | More general and more powerful. |
| Gives total current first, then works backward to branch values. | Can solve for multiple unknown currents directly. |
Side-by-Side Example: Same Circuit, Two Methods
A \(12\ \text{V}\) battery is connected to a \(2\ \Omega\) resistor and a \(4\ \Omega\) resistor in series.
Equivalent Resistance
\[ R_{\text{eq}}=2+4=6\ \Omega \] \[ I=\frac{V}{R_{\text{eq}}}=\frac{12}{6}=2.0\ \text{A} \]
Kirchhoff Loop
Move around the loop: \[ +12-2I-4I=0 \] \[ 12=6I \] \[ I=2.0\ \text{A} \]
Side-by-Side Example: Series–Parallel Circuit
A \(12\ \text{V}\) battery is connected to a \(4\ \Omega\) resistor in series with two \(6\ \Omega\) resistors in parallel.
Equivalent Resistance Method
Parallel pair: \[ \frac{1}{R_p}=\frac{1}{6}+\frac{1}{6}=\frac{1}{3} \] \[ R_p=3\ \Omega \] Total: \[ R_{\text{eq}}=4+3=7\ \Omega \] \[ I_{\text{total}}=\frac{12}{7}=1.71\ \text{A} \]
Kirchhoff Reasoning
Let total current be \(I\), and let the branch currents be \(I_2\) and \(I_3\). Since the parallel resistors are equal: \[ I_2=I_3=\frac{I}{2} \] Loop through either branch: \[ +12-4I-6\left(\frac{I}{2}\right)=0 \] \[ 12-4I-3I=0 \] \[ I=\frac{12}{7}=1.71\ \text{A} \]
Teacher Note / Student Strategy
Use equivalent resistance when the structure is obvious. Use Kirchhoff’s Laws when there are multiple loops, more than one battery, or no clean way to collapse the circuit. For AP-level work, students should understand both methods as expressions of the same conservation principles.
11. Practice Problems
Problem 1: Electric Force
Two charges, \(+4.0\ \mu\text{C}\) and \(+5.0\ \mu\text{C}\), are separated by \(0.30\ \text{m}\). Find the electric force and state whether it is attractive or repulsive.
Show Solution
\[ F=k\frac{|q_1q_2|}{r^2} =(8.99\times10^9)\frac{(4.0\times10^{-6})(5.0\times10^{-6})}{(0.30)^2} \] \[ F\approx 2.0\ \text{N} \] Both charges are positive, so the force is repulsive.
Problem 2: Voltage
A device transfers \(240\ \text{J}\) of energy as \(20\ \text{C}\) of charge passes through it. Find the potential difference.
Show Solution
\[ V=\frac{W}{q}=\frac{240}{20}=12\ \text{V} \]
Problem 3: Capacitance
A \(470\ \mu\text{F}\) capacitor is connected to a \(9.0\ \text{V}\) battery. Find the charge stored.
Show Solution
Convert: \[ 470\ \mu\text{F}=470\times10^{-6}\ \text{F} \] \[ Q=CV=(470\times10^{-6})(9.0)=4.23\times10^{-3}\ \text{C} \]
Problem 4: Series Circuit
A \(9.0\ \text{V}\) battery is connected to \(3\ \Omega\), \(6\ \Omega\), and \(9\ \Omega\) resistors in series. Find total resistance and current.
Show Solution
\[ R_{\text{eq}}=3+6+9=18\ \Omega \] \[ I=\frac{9.0}{18}=0.50\ \text{A} \]
Problem 5: Parallel Circuit
A \(12\ \text{V}\) battery is connected to \(4\ \Omega\) and \(12\ \Omega\) resistors in parallel. Find equivalent resistance and total current.
Show Solution
\[ \frac{1}{R_{\text{eq}}}=\frac{1}{4}+\frac{1}{12}=\frac{3}{12}+\frac{1}{12}=\frac{4}{12} \] \[ R_{\text{eq}}=3\ \Omega \] \[ I_{\text{total}}=\frac{12}{3}=4.0\ \text{A} \]
Problem 6: Series–Parallel Circuit
A \(10\ \Omega\) resistor is in series with a parallel combination of \(20\ \Omega\) and \(30\ \Omega\). The battery is \(24\ \text{V}\). Find total resistance and total current.
Show Solution
Parallel section: \[ \frac{1}{R_p}=\frac{1}{20}+\frac{1}{30}=\frac{3}{60}+\frac{2}{60}=\frac{5}{60} \] \[ R_p=12\ \Omega \] Total: \[ R_{\text{eq}}=10+12=22\ \Omega \] \[ I=\frac{24}{22}=1.09\ \text{A} \]
Problem 7: Kirchhoff Loop
A single loop contains a \(15\ \text{V}\) battery and two resistors, \(5\ \Omega\) and \(10\ \Omega\). Write the Kirchhoff loop equation and solve for current.
Show Solution
\[ +15-5I-10I=0 \] \[ 15=15I \] \[ I=1.0\ \text{A} \]
12. Closing Connection
Electrostatics and circuits are not separate topics. Electrostatics explains why charges exert forces and why separated charge stores energy. Circuits explain what happens when charges are given a continuous path and a maintained potential difference. The same ideas — charge, force, field, energy, and conservation — connect the spark from a Van de Graaff generator to the current flowing through a resistor network.
HS-PS3 HS-PS3-5 Planning Investigations Analyzing Data PS3.B / PS3.C Systems & Models
Circuits Exploration Inquiry
A guided investigation of charge movement, voltage, current, resistance, capacitance, and circuit design using simulation, prediction, measurement, and physical construction.
1. Built-In Circuit Simulator
Use the simulator below to plan your circuit before building it physically. The goal is not to “play until it works.” The goal is to create a prediction, test the prediction, and then explain any differences between your model and your measurements.
If the embedded simulator does not load on your device, use the “Open Full Simulator” button. Your work should still include a circuit diagram, predictions, and measured values.
How to Use the Simulator Productively
- Sketch the circuit first using standard symbols.
- Calculate the expected equivalent resistance, voltage, current, or time behavior.
- Build the circuit in the simulator.
- Use meters or visual current flow to test your prediction.
- Revise your diagram if the simulator reveals a mistake.
- Only then build the physical circuit.
2. Safety Expectations
- Use power supplies at \(9\text{ V}\) or less unless instructed otherwise.
- Check component tolerances before powering bulbs, LEDs, capacitors, and resistors.
- Design and predict on paper or in the simulator before powering a circuit.
- Turn off and disconnect power before modifying any physical circuit.
- Keep fingers out of live circuits when measuring with a multimeter.
- Remember that capacitors can hold charge after the circuit is off.
- Discharge capacitors through a resistor, not through your body or a direct short.
- Do not leave circuits powered longer than necessary.
- Observe capacitor polarity carefully. Incorrect polarity can cause failure or popping.
3. Learning Goals
Physics Goals
- Explain charge movement using electric potential difference.
- Use Ohm’s Law to predict voltage, current, and resistance.
- Compare series, parallel, and series–parallel behavior.
- Use capacitor charging and discharging behavior to estimate \(RC\) time.
- Connect circuit behavior back to electrostatics.
Inquiry Goals
- Make predictions before building.
- Create accurate circuit diagrams.
- Measure with a multimeter.
- Photograph and annotate physical builds.
- Use evidence to revise explanations.
- Write a clear technical report.
4. Required Inquiry Workflow
Each challenge must follow the same cycle. This is the heart of the assignment.
Predict → Simulate → Build → Measure → Explain
- Predict: Draw the circuit and calculate expected values.
- Simulate: Build the circuit in the simulator and compare with your prediction.
- Build: Build the physical circuit safely.
- Measure: Use the multimeter to collect voltage, current, or resistance data.
- Explain: Use equations and concepts to explain what happened.
5. Circuit Challenges
Challenge 1 — Ohm’s Law and Equivalent Resistance
Create circuits with varying arrangements of one, two, three, four, or more resistors. Use these circuits to verify equations for equivalent resistance.
Conceptual Check
Why does adding resistors in series increase equivalent resistance while adding branches in parallel decreases equivalent resistance?
Calculation Check
A \(3\ \Omega\) resistor and a \(6\ \Omega\) resistor are connected in parallel. That parallel combination is then placed in series with a \(4\ \Omega\) resistor. Find the equivalent resistance.
Evidence to Include
- At least three resistor arrangements.
- Predicted equivalent resistance for each.
- Simulator screenshot or exported circuit evidence.
- Multimeter measurements.
- A short paragraph explaining agreement or disagreement.
Challenge 2 — Step-Down Voltage Divider
Create a circuit that reduces a \(9\text{ V}\) source to about \(3\text{ V}\) across one part of the circuit.
Conceptual Check
Why does a voltage divider require at least two resistive elements rather than just one resistor connected to a battery?
Calculation Check
Choose resistor values that would make \(V_{\text{out}}\approx3\text{ V}\) from a \(9\text{ V}\) source. Show your calculation.
Evidence to Include
- Diagram labeling \(R_1\), \(R_2\), \(V_{\text{in}}\), and \(V_{\text{out}}\).
- Predicted output voltage.
- Simulator reading.
- Measured physical circuit output.
Challenge 3 — Mini-Bulb with Switch and Button
Add a mini-bulb to your circuit and wire it through a switch. Then rewire it through a button.
Conceptual Check
What is the difference between an open circuit and a closed circuit? Why does the bulb turn off when the switch opens?
Calculation Check
If the bulb is rated for \(3\text{ V}\) and draws \(0.20\text{ A}\), estimate its operating resistance using Ohm’s Law.
Evidence to Include
- Diagram with switch.
- Diagram with button.
- Photo of the working circuit.
- Explanation of how the switch changes the circuit path.
Challenge 4 — RC Time: Charging and Discharging a Capacitor
Charge a capacitor and then discharge it through a resistor while measuring voltage over time. Compare your observed discharge behavior with the expected \(RC\) time scale.
Conceptual Check
Why does the capacitor voltage decrease quickly at first and then more slowly as time passes?
Calculation Check
A \(1000\ \mu\text{F}\) capacitor discharges through a \(2.2\text{ k}\Omega\) resistor. Calculate \(\tau\) and estimate \(5RC\).
Evidence to Include
- Charging circuit diagram.
- Discharging circuit diagram.
- Table of voltage versus time.
- Graph of voltage versus time.
- Comparison of observed discharge time to \(5RC\).
Challenge 5 — Series and Parallel Bulbs Powered by a Capacitor
Charge a large capacitor and use it to power three mini-bulbs arranged in series and then in parallel. Compare brightness and duration.
Conceptual Check
Why might bulbs in parallel appear brighter than bulbs in series when connected to the same source?
Calculation Check
If three identical bulbs each have resistance \(6\ \Omega\), find the equivalent resistance when all three are in series and when all three are in parallel.
Evidence to Include
- Series bulb diagram.
- Parallel bulb diagram.
- Photos or observations comparing brightness.
- Equivalent resistance calculations.
- Explanation using current, voltage, and power.
Challenge 6 — Switchable Capacitor Charge/Discharge Circuit
Create a circuit that charges a capacitor, then switches to discharge the capacitor through a light bulb without burning out the bulb.
Conceptual Check
Why must you consider capacitor polarity and bulb ratings before powering the circuit?
Calculation Check
A \(2200\ \mu\text{F}\) capacitor is charged to \(9\text{ V}\). Calculate the energy stored in the capacitor.
Evidence to Include
- Diagram showing the charging state.
- Diagram showing the discharging state.
- Photo of the final circuit.
- Explanation of current direction during charge and discharge.
- Safety explanation for protecting the bulb and capacitor.
6. Final Report Requirements
Your final report should read like an organized technical narrative, not a list of answers. It should explain what you tried, what you predicted, what happened, and how physics explains your observations.
Opening Concept Page
Begin your report with an illustrated explanation of the following concepts:
- Separation of charge
- The Van de Graaff generator
- Charging by induction
- Voltage
- Current
For each concept, include both an illustration and an explanation connecting the concept to forces, energy, work, power, or charge movement.
Required Report Sections for Each Challenge
- Goal: What were you trying to create, test, or verify?
- Prediction: What did you calculate or expect before building?
- Diagram: Include a clean circuit diagram with labels.
- Simulation: Explain what happened in the simulator.
- Physical Build: Include a photo of the completed physical circuit.
- Measurements: Include voltage, current, resistance, time, or brightness observations as appropriate.
- Analysis: Tie observations back to equations and concepts.
- Revision: Explain what changed between your first attempt and final circuit.
7. Rubric
Constructing Explanations Analyzing Data Systems & Models
The main circuit challenges are scored on the quality of the circuit, modeling, explanation, and depth of inquiry. The final report is scored separately for evidence, organization, and clarity.
| Score | Description |
|---|---|
| 1 pt | Attempted to build or model the circuit but did not complete the task successfully. |
| 2 pts | Completed a working circuit, but modeling, diagrams, or explanations are missing or insufficient. |
| 3 pts | Completed a working circuit and attempted to use theory or mathematics to model the result. |
| 4 pts | Circuit and mathematical modeling are accurate and clearly connected to observations. |
| 5 pts | Completed accurate work and asked or investigated additional meaningful questions. |
Challenge Categories
- Ohm’s Law and equivalent resistance circuits
- Step-down voltage divider
- Mini-bulb with switch and button
- RC time charging and discharging
- Series and parallel bulbs
- Switchable capacitor charge/discharge circuit
Final Report: 25 Points
| Category | Points | Expectation |
|---|---|---|
| Photos | 5 | All required circuit photos are present and clearly connected to the written discussion. |
| Diagrams | 5 | All required circuit diagrams are present, labeled, and accurate. |
| Format | 5 | Report is easy to follow and formatted as a complete technical document. |
| Clarity of Explanations | 5 | Explanations clearly connect observations to physics concepts, equations, and measurements. |
| Organization | 5 | Report is logically ordered and shows the progression from prediction to evidence to conclusion. |
8. Final Reflection
Reflection Prompts
- Which circuit was easiest to predict? Why?
- Which circuit behaved most differently from your first prediction?
- What did the simulator help you understand before building physically?
- Where did real-world behavior differ from the ideal model?
- How do voltage and current connect back to electrostatics and movement of charge?