Electric charge (symbol: q or Q) is a fundamental property of matter that causes it to experience a force near other charged matter. Two types: positive (+) and negative (−).
Think of charge as a "tag" on particles. Protons carry +, electrons carry −, neutrons are neutral (no tag).
The Golden Rule
+
⟵ REPEL ⟶
+
−
⟵ REPEL ⟶
−
+
⟶ ATTRACT ⟵
−
"Opposites attract, likes repel." This shows up on EVERY test.
Carpet shuffle → ~10–100 μC on your body. Small but enough to feel a zap!
Lightning bolt → ~5 C transferred in milliseconds. Enormous!
2. The Atom — Where Charge Lives
Schematic atom (Lithium-7: 3 protons, 4 neutrons, 3 electrons) · Protons & neutrons locked in the nucleus · Electrons orbit outside and can move
Particle
Charge
Location
Can Move?
Proton
+1.602 × 10⁻¹⁹ C
Nucleus
❌ No
Neutron
0
Nucleus
❌ No
Electron
−1.602 × 10⁻¹⁹ C
Orbiting
✅ Yes!
Key Insight
In solid metals, only electrons move. "Positive charge" means it lost electrons. "Negative charge" means it gained electrons. (In liquids like salt water, charge is carried by ions — both positive and negative — not electrons.)
Never say something "gained positive charge." It lost electrons. Protons never leave the nucleus!
3. Three Properties of Electric Charge
① Quantization
Rule
Charge comes in discrete packets — always a whole-number multiple of e = 1.602 × 10−19 C.
q = n × e (n is any integer)
Example 1
Q: Is 4.806 × 10⁻¹⁹ C a valid charge?
n = 4.806×10⁻¹⁹ ÷ 1.602×10⁻¹⁹ = 3 ✅ Whole number → Valid!
Example 2
Q: Is 2.5 × 10⁻¹⁹ C a valid charge?
n = 2.5×10⁻¹⁹ ÷ 1.602×10⁻¹⁹ = 1.56 ❌ Not whole → Impossible!
② Conservation
Rule
Charge cannot be created or destroyed. Total charge in an isolated system is constant. Charge can only be transferred.
Most metals = conductors, most non-metals = insulators. Key exceptions: graphite (carbon) conducts electricity, semiconductors (silicon, germanium) conduct partially, and ionic solutions (salt water) conduct via ions.
Example
Q: Why do electricians wear rubber gloves?
Rubber is an insulator → blocks electron flow → prevents electric shock.
5. Static Electricity — Charges at Rest
Definition
Static electricity = buildup of electric charge on an object's surface. Unlike current electricity (flowing electrons), static charges stay put until they find a path to discharge.
Touching a charged object to a neutral one shares charge directly.
📌 Charged rod touches metal sphere → both share charge.
③ Induction (No Contact!)
A charged object near a conductor rearranges charges inside without touching.
📌 Negative rod near metal sphere → electrons repelled to far side → near side (+), far side (−). Ground the far side → electrons escape to ground → remove rod → sphere is permanently (+)!
Induction: charge an object without ever touching it
Sources & Hazards of Static
Sources: Friction (walking on carpet, clothes dryer), separation of materials, flowing liquids/gases
⚡ Sparks: Can ignite flammable vapors, dusts, or gases (gas station danger!)
💻 ESD: Electrostatic discharge destroys sensitive electronics like computer chips
⚡ Lightning: Massive static discharge between clouds and ground
The #1 tested static hazard: "sparks that can ignite flammable vapors." That's why gas stations have grounding straps and "touch metal before refueling" signs.
6. Electric Fields — The Invisible Force Zone
Definition
An electric field (E) is the region around a charged object where other charges experience a force. Every charge creates a field extending into space.
Field Line Rules (Important!)
Lines point away from (+) charges and toward (−) charges
More lines = stronger field
Lines never cross
Lines are closer together where the field is stronger
Electric field lines: outward from positive, inward toward negative
Electric Field Formula
E = F / q
E = electric field strength (N/C or V/m)F = force on the test charge (Newtons)q = test charge (Coulombs)
You can also calculate the field from a single point charge:
E = ke × Q / r²
ke = 8.99 × 10⁹ N·m²/C² (Coulomb's constant)Q = source charge creating the fieldr = distance from the charge
Example
Q: A +2 μC charge creates an electric field. What is E at a point 0.5 m away?
E = k × Q / r² = (8.99 × 10⁹)(2 × 10⁻⁶) / (0.5)²
E = (8.99 × 10⁹)(2 × 10⁻⁶) / 0.25 = 17,980 / 0.25 = 71,920 N/C
Direction: away from the positive charge.
Think of it like gravity: A planet creates a gravitational field around it. Similarly, a charge creates an electric field around it. Other charges placed in that field feel a force — just like objects near a planet feel gravitational pull!
7. Coulomb's Law — The Math of Electric Force
The Big Formula
This is one of the most important equations for Circuit Lab. It tells you the force between two charged objects.
F = ke × |q₁| × |q₂| / r²
F = electrostatic force between the charges (Newtons, N)ke = Coulomb's constant = 8.99 × 10⁹ N·m²/C²q₁, q₂ = the two charges (Coulombs, C)r = distance between the centers of the charges (meters, m)
What Coulomb's Law Tells Us
Directly Proportional to Charges
Double one charge → force doubles. Double both charges → force quadruples (4×).
Inversely Proportional to Distance²
Double the distance → force drops to 1/4. Triple the distance → force drops to 1/9. This is called an inverse-square law.
Worked Example 1
Q: Two charges, q₁ = +3 μC and q₂ = −5 μC, are 0.2 m apart. What is the force?
Given: q₁ = 3 × 10⁻⁶ C, q₂ = 5 × 10⁻⁶ C, r = 0.2 m
F = k × |q₁| × |q₂| / r²
F = (8.99 × 10⁹)(3 × 10⁻⁶)(5 × 10⁻⁶) / (0.2)²
F = (8.99 × 10⁹)(15 × 10⁻¹²) / 0.04
F = 0.13485 / 0.04 = 3.37 N Direction: Attractive (opposite charges → they pull toward each other).
Worked Example 2 — Distance Change
Q: Two charges have a force of 12 N between them. If you double the distance, what's the new force?
Inverse square law: F ∝ 1/r²
Double distance → r becomes 2r → r² becomes 4r²
New force = 12 / 4 = 3 N
The forcedrops to one-quarter!
Worked Example 3 — Charge Change
Q: Two charges produce a force of 8 N. If you triple one of the charges, what happens to the force?
F ∝ q₁ × q₂ (directly proportional)
Triple one charge → force triples → 24 N
Watch out for the r² in the denominator! Students often forget to square the distance. If r = 0.3 m, you use 0.09, not 0.3.
Coulomb's Law looks just like Newton's Law of Gravity!
Gravity: F = G × m₁m₂/r² (masses attract)
Coulomb: F = k × q₁q₂/r² (charges attract OR repel)
Same shape, but gravity only attracts. Electric force can attract or repel!
8. Capacitance — Storing Charge
Definition
Capacitance is the ability to store electric charge. A capacitor is a device built specifically to do this — two metal plates separated by an insulating material (dielectric).
C = Q / V
C = capacitance (Farads, F)Q = charge stored (Coulombs, C)V = voltage across the capacitor (Volts, V)
1 Farad is HUGE — it means storing 1 coulomb for every volt. Real capacitors are usually measured in:
Prefix
Symbol
Value
Common Use
Microfarad
μF
10⁻⁶ F
Power supplies, audio circuits
Nanofarad
nF
10⁻⁹ F
Signal filtering
Picofarad
pF
10⁻¹² F
Radio, high-frequency circuits
How a Capacitor Works
Two metal plates separated by an insulator. Charge builds up on the plates.
What Affects Capacitance?
Larger plate area → more capacitance (more room for charge)
Smaller plate gap → more capacitance (stronger electric field)
Better dielectric → more capacitance (insulator quality matters)
Worked Example
Q: A 100 μF capacitor is connected to a 9V battery. How much charge does it store?
C = Q / V → Q = C × V Q = (100 × 10⁻⁶)(9) = 900 μC = 0.0009 C
Worked Example
Q: A capacitor stores 0.005 C at 25V. What is its capacitance?
C = Q / V = 0.005 / 25 = 0.0002 F = 200 μF
Polarized vs. Non-Polarized Capacitors
Polarized (Electrolytic)
Has a + and − side. Must be connected correctly or it can be damaged/explode!
Examples: aluminum electrolytic, tantalum. Typical values: larger (1 μF to thousands of μF)
Non-Polarized (Ceramic/Film)
Can be connected either way — no polarity.
Examples: ceramic disc, film capacitors. Typical values: smaller (pF to low μF)
On the test, if they show a capacitor with a + marking, it's polarized. Connecting it backward can destroy it!
Think of a capacitor like a tiny rechargeable battery — but one that charges and discharges in fractions of a second. It stores energy temporarily, not long-term.
Massive charge buildup in clouds discharges to ground
Lightning rods (conductors that safely direct current to ground)
Surgical/hospital risks
Sparks near flammable anesthetic gases
Conductive shoes, grounded equipment
Key Takeaway
The common thread: static discharge (sparks) + flammable material = danger. Prevention always involves grounding — giving charge a safe path to flow away.
10. Key Person: Charles-Augustin de Coulomb
Historical Figure
Charles-Augustin de Coulomb (1736–1806) was a French physicist who discovered the law governing the force between electric charges.
Nationality: French
Era: 18th century
Famous for: Coulomb's Law — the inverse-square law of electrostatic force
Tool: Invented the torsion balance to precisely measure tiny forces between charged objects
Unit named after him: The coulomb (C) — the SI unit of electric charge
Key insight: Showed that electric force follows the same mathematical pattern as Newton's gravitational force (inverse-square law)
The test will ask which scientist is associated with the force between charges or the unit of charge. Answer: Coulomb. Don't confuse with Ampere (current), Ohm (resistance), or Volta (voltage).
11. Practice Problems (15 Questions)
Select an answer or type your response, then click Submit to check yourself.
Q1
Which of the following best describes Coulomb's Law?
Q2
Which unit measures electric charge?
Q3
Which of the following best represents a hazard of static electricity?
Q4
An object has 5 more protons than electrons. What is its charge?
Q5
Two charges of +4 μC and −6 μC are 0.3 m apart. Calculate the force between them.
Q6
Two charges exert a force of 20 N on each other. If the distance between them is tripled, what is the new force?
Q7
Define capacitance and give its SI unit.
Q8
A rubber rod is rubbed with fur. The rod becomes negatively charged. Explain what happened using conservation of charge.
Q9
Electric field lines around a positive charge point:
Q10
A 220 μF capacitor is connected to a 12V battery. How much charge is stored?
Q11
Is a charge of 6.408 × 10⁻¹⁹ C possible? Why or why not?
Q12
Which of the following is a conductor?
Q13
Two identical metal spheres have charges of +8 μC and −2 μC. They are touched together and then separated. What is the charge on each sphere?
Q14
Two charges produce a force of 10 N. If one charge is doubled AND the distance is halved, what is the new force?
Q15
A capacitor stores 500 μC of charge when connected to a 10V source. What is its capacitance? If the voltage is increased to 20V, how much charge will it store?
12. 📝 Cheat Sheet — Topic 1
Concept
Key Facts
Electric Charge
Property of matter. Two types: + and −. Like repels, opposite attracts. Unit: Coulomb (C).
Elementary Charge
e = 1.602 × 10⁻¹⁹ C. Proton = +e, electron = −e.
Quantization
q = n × e (charge is always a whole number of electrons).
Conservation
Charge cannot be created or destroyed — only transferred.
Conductors
Metals. Electrons flow freely. Low resistance.
Insulators
Non-metals (rubber, glass, plastic). Block electron flow.
Static Electricity
Charge buildup. Three methods: friction, conduction, induction.
Static Hazards
#1: Sparks igniting flammable vapors. Also ESD on electronics.
Electric Field
E = F/q = k×Q/r². Lines: away from (+), toward (−). Unit: N/C or V/m.
Coulomb's Law
F = k|q₁||q₂|/r². k = 8.99 × 10⁹. Inverse-square law.
Capacitance
C = Q/V. Unit: Farad (F). μF, nF, pF for smaller values.
Coulomb (person)
French physicist (1736–1806). Torsion balance. Inverse-square law for charge.
Formula Quick Reference:
F = k|q₁||q₂|/r² (force between charges)
E = F/q = kQ/r² (electric field)
C = Q/V (capacitance)
k = 8.99 × 10⁹ N·m²/C² | e = 1.602 × 10⁻¹⁹ C
📺 Video References — Electric Charge & Field
Curated Khan Academy and YouTube videos for middle school Science Olympiad prep. Watch alongside the study guide for deeper understanding.
Rub a balloon on a sweater and explore static electricity, charge transfer, and attraction/repulsion. Free from University of Colorado.
🎮 Interactive — the best way to "see" static electricity in action
💡 Study tip: Watch the middle school videos first (3–8 min each), then tackle the deeper Physics ones (10–22 min) for competition-level prep. Use the PhET simulator to experiment yourself!
1. What Is Direct Current (DC)?
Key Definition
Direct Current (DC) is electric current that flows in one direction only. Electrons always move from the negative terminal, through the circuit, and back to the positive terminal.
Think of DC like a river — water flows steadily in one direction. The current never reverses.
Characteristics of DC
Direction: Always one way (unidirectional)
Voltage: Constant (steady) — doesn't oscillate
Frequency: 0 Hz (no oscillation)
Graph shape: A flat horizontal line on a voltage-vs-time graph
DC voltage stays flat — constant and steady, unlike AC which oscillates
"DC = Direct = one Direction, Constant." Just remember: DC is the simple one. One way, steady flow.
2. DC vs. AC — Quick Comparison
Why This Matters
DC and AC are the two fundamental types of electric current. This topic focuses on DC, but know the difference!
Property
DC (Direct Current)
AC (Alternating Current)
Direction
One way only
Reverses back and forth
Voltage
Constant
Oscillates (sine wave)
Frequency
0 Hz
60 Hz (US) / 50 Hz (world)
Source
Batteries, solar cells
Power plants, wall outlets
Travel distance
Short (loses energy)
Long (transformers help)
Symbol
— (straight line)
~ (wavy line)
The test loves: "Name a DC device" → flashlight, phone, laptop. "Name an AC device" → wall outlet, toaster, refrigerator.
3. DC Sources — Batteries & Beyond
Key Concept
A DC source provides a steady voltage that pushes electrons in one direction. The most common DC source is a battery.
Source
How It Works
Example
Battery (cell)
Chemical energy → electrical energy
AA, AAA, 9V, car battery
Solar cell
Sunlight knocks electrons loose in semiconductor
Roof panels, calculators
DC power supply
Converts AC from wall → DC (rectifier)
Phone charger, laptop adapter
DC generator
Spinning coil in magnetic field + commutator
Old-style generators
💡 Real World
Your phone charger takes AC from the wall (120V, 60Hz in the US) and converts it to DC (usually 5V). That brick on the cable is a rectifier!
A battery does NOT "store electricity." It stores chemical energy and converts it to electrical energy through chemical reactions.
4. Inside a Battery — How It Works
Key Components
A battery has three essential parts: an anode (negative electrode), a cathode (positive electrode), and an electrolyte (chemical medium).
Cations (+) migrate toward the cathode; anions (−) migrate toward the anode. Electrons flow through the external circuit from anode to cathode.
Ions, Anions & Cations
Key Definitions
An ion is an atom (or group of atoms) that has gained or lost electrons, giving it a net charge.
Cation (+) = a positive ion (lost electrons). Think: cation has a "+" in the "t". Example: Na⁺, Cu²⁺
Anion (−) = a negative ion (gained electrons). Think: a negative ion. Example: Cl⁻, SO₄²⁻
Memory trick for ion flow:Cations go to the Cathode. Anions go to the Anode. The letters match!
How a Battery Works
Anode (−): Chemical reaction oxidizes → releases electrons into the external circuit
Electrons flow through the external circuit from anode → cathode (this is your current!)
Cathode (+): Chemical reaction reduces → absorbs electrons from the circuit
Inside the electrolyte:Cations (+) migrate toward the cathode, and anions (−) migrate toward the anode — completing the circuit internally
When the chemicals are used up → battery is "dead"
Don't confuse conventional current direction with electron flow! Conventional current flows from the + terminal (cathode) through the external circuit to the − terminal (anode). But electrons actually move the opposite way: from anode (−) through the external circuit to cathode (+).
Cell vs. Battery
A cell = single electrochemical unit (one AA). A battery = two or more cells connected (a 9V = six 1.5V cells in series). Everyday language uses "battery" for both.
"An Ox, Red Cat" — Anode = Oxidation, Reduction = Cathode. Works in chemistry AND physics!
5. Battery Configurations: Series & Parallel
Key Concept
Connect multiple batteries together to change total voltage or capacity. Series = voltages add. Parallel = capacity adds.
Batteries in Series
Connect + to − in a chain. Voltages add up. Current capacity stays the same.
Vtotal = V₁ + V₂ + V₃ + ...
Series: connect (+) to (−) of the next battery. Voltage adds, capacity stays the same.
Batteries in Parallel
Connect all (+) together and all (−) together. Voltage stays the same, capacity multiplies.
Vtotal = V (same as one battery)
Parallel: voltage unchanged, but the batteries last longer (capacity adds).
Science Olympiad often asks: "How do you increase voltage?" → Series. "How do you increase battery life?" → Parallel.
Never connect batteries of different voltages in parallel! The higher-voltage battery will force current backward through the weaker one, causing overheating or damage.
6. Internal Resistance
Key Concept
Every real battery has some resistance inside it called internal resistance (r). It wastes a bit of voltage as heat inside the battery, so the voltage you actually get is less than the rated voltage.
Vterminal = EMF − I × r
Vterminal = voltage actually delivered to the circuit (V)EMF (ε) = electromotive force — the battery's "ideal" voltage (V)I = current flowing (A)r = internal resistance (Ω)
Worked Example
Q: A 9V battery has an internal resistance of 0.5 Ω. If 2A of current flows, what is the terminal voltage?
Vterminal = EMF − Ir = 9 − (2)(0.5) = 9 − 1 = 8V
The battery "loses" 1V as heat internally, delivering only 8V to the circuit.
Worked Example 2 — Finding Internal Resistance
Q: A battery's EMF is 12V. When a 4A current flows, the terminal voltage drops to 10V. What is the internal resistance?
Vterminal = EMF − Ir
10 = 12 − 4r → 4r = 2 → r = 0.5 Ω
Old batteries have higher internal resistance! That's why a "dead" battery still shows ~1.3V on a meter but can't power your flashlight — too much voltage is dropped internally when current flows.
If a test shows a battery with EMF and internal resistance, always use V = EMF − Ir for the actual circuit voltage, not the rated EMF.
7. DC Circuit Components
A DC circuit is made of components connected by wires. Know each component's job!
🔗 Wires
Conductors (usually copper) that carry current between components. Ideal wires have zero resistance. In reality, very low resistance.
🔲 Resistors
Oppose current flow. Measured in Ohms (Ω). Convert electrical energy to heat. Used to control current and voltage in circuits.
🔘 Switches
Open or close a circuit. Closed switch = current flows. Open switch = circuit broken, no current. Like a door for electrons.
💡 Bulbs (Incandescent)
Resistors that produce light and heat. The filament has resistance. Brightness depends on power (P = I²R). More current → brighter light.
💡 LEDs (Light-Emitting Diodes)
Emit light when current flows in the correct direction only. More efficient than bulbs. Require a current-limiting resistor to prevent burnout. Have a minimum forward voltage (~1.8–3.3V).
⚡ Fuses
Safety devices. A thin wire that melts and breaks if current exceeds a safe limit. One-time use — must be replaced. Protects the circuit from overcurrent damage.
Component
Symbol Description
Key Property
Battery/Cell
Long line (+) and short line (−)
Provides EMF (voltage)
Resistor
Rectangle or zigzag
Resistance in Ω
Switch (open)
Gap in wire with lever
Breaks circuit
Switch (closed)
Lever touching contact
Completes circuit
Bulb/Lamp
Circle with X or filament
Light + heat output
LED
Triangle with arrow pointing to line
Light, one direction only
Fuse
Rectangle with line through it
Current limit protection
Capacitor
Two parallel lines
Stores charge (Farads)
For Science Olympiad: know whether each component is in series (one path) or can be in parallel (multiple paths). Fuses are always in series — to protect the whole circuit.
8. Circuit Diagrams & Schematic Symbols
Why Schematics Matter
A schematic diagram uses standardized symbols to represent circuits. Every engineer and physicist uses the same symbols worldwide. Reading schematics is a core Science Olympiad skill!
Memorize these symbols — every schematic question uses them!
Key rule: A dot (•) where wires cross means they ARE connected. No dot means they just cross without connecting. This trips up many students!
How to Read a Schematic
Identify the power source (battery)
Trace the path(s) electrons can take
Identify each component and its value
Determine if components are in series (one path) or parallel (multiple paths)
Apply Ohm's Law and Kirchhoff's laws
9. Series Circuits
Key Definition
In a series circuit, components are connected end-to-end in a single loop. There is only one path for current to flow.
Series circuit: one loop, same current through all components
Rules for Series Circuits
Current — Same Everywhere
Itotal = I₁ = I₂ = I₃ Current has only one path, so the same current flows through every component.
Voltage — Divides Up
Vtotal = V₁ + V₂ + V₃ Each resistor "uses up" part of the total voltage. They add back to the source voltage.
Resistance — Adds Directly
Rtotal = R₁ + R₂ + R₃ + ... More resistors in series = more total resistance.
If One Component Fails...
The circuit breaks completely. Like old Christmas lights — if one bulb goes out, they all go out. No alternative path!
Worked Example
Q: Three resistors R₁ = 10Ω, R₂ = 20Ω, R₃ = 30Ω are in series with a 12V battery. Find total resistance, current, and voltage across each resistor.
Students often try to ADD resistances in parallel — wrong! Use the reciprocal formula. Rtotal gets SMALLER when you add parallel resistors.
Special case — Two equal resistors in parallel: Rtotal = R/2. Three equal resistors: R/3. This shortcut saves time on timed tests!
11. Series vs. Parallel — Side by Side
Property
Series
Parallel
Number of paths
One path only
Multiple paths
Current
Same through all: I₁=I₂=I₃
Splits: Itotal=I₁+I₂+I₃
Voltage
Divides: Vtotal=V₁+V₂+V₃
Same across all: V₁=V₂=V₃
Resistance formula
Rtotal = R₁+R₂+R₃ (increases)
1/Rtotal=1/R₁+1/R₂+1/R₃ (decreases)
Effect of adding more
R increases, current decreases
R decreases, current increases
One component fails
All stop (circuit broken)
Others keep working
Bulb brightness
Dimmer (share voltage)
Same brightness (each gets full V)
Real-world use
Old string lights, simple switches
House wiring, modern lights
Bulb Brightness in Series vs. Parallel
Series — Dimmer Bulbs
Two 100Ω bulbs in series with 12V: I = 12/(100+100) = 0.06A Power per bulb: P = I²R = (0.06)² × 100 = 0.36W Bulbs are dim — they share the voltage.
Parallel — Same Brightness
Two 100Ω bulbs in parallel with 12V: Each sees full 12V. I = 12/100 = 0.12A each. Power per bulb: P = V²/R = 144/100 = 1.44W Bulbs are bright — each gets full voltage.
Key exam question: "If you add another bulb in parallel, what happens to the others?" → They stay the same brightness (same voltage, same power). "If you add another bulb in series?" → They all get dimmer (voltage divides further).
12. Ohm's Law & Power
The Most Important Formula in DC Circuits
Ohm's Law: V = I × R | I = V/R | R = V/I
V = I × R
V = Voltage (Volts, V) — the "push" driving currentI = Current (Amperes, A) — the rate of electron flowR = Resistance (Ohms, Ω) — opposition to current flow
The Ohm's Law Triangle
Cover V → I×R. Cover I → V/R. Cover R → V/I.
Power Formulas
Power (P) measures how fast electrical energy is converted. Unit: Watts (W).
P = I × V = I²R = V²/R
P = Power in Watts (W)P = IV — use when you know current and voltageP = I²R — use when you know current and resistanceP = V²/R — use when you know voltage and resistance
All the Ohm/Power Formulas
From V=IR and P=IV, you can derive 12 combinations:
V = IR = P/I = √(PR)
I = V/R = P/V = √(P/R)
R = V/I = V²/P = P/I²
P = IV = I²R = V²/R
Energy vs. Power
Power = energy per second (Watts) Energy = Power × Time
E = P × t (Joules when P in W, t in seconds)
Don't confuse voltage and power! A 60W bulb and a 100W bulb connected to the same 120V outlet: the 100W bulb draws more current (I = P/V = 0.83A vs 0.5A) and is brighter.
13. DC Hazards & Safety
Why It Matters
DC electricity can be dangerous. Understanding hazards keeps you and circuits safe — and it's tested on Science Olympiad!
Electric Shock
Current through the human body is what kills — not voltage alone. As little as 100 mA (0.1A) through the heart can be fatal.
Current
Effect on Human Body
1–5 mA
Tingling, slight shock
10–20 mA
Painful shock, cannot let go
50–100 mA
Severe shock, heart risk
100–200 mA
Ventricular fibrillation — potentially fatal
Over 1A
Severe burns, cardiac arrest
Short Circuits
What Is a Short Circuit?
A direct connection between + and − with very low resistance. Current follows an unintended low-resistance path.
Using R = V/I: if R ≈ 0, then I → huge → wires overheat → fire!
Prevention
Fuses — melt to break the circuit
Circuit breakers — reset automatically
Proper insulation on all wires
Don't connect battery terminals directly!
Overheating & Thermal Runaway
Components dissipate power as heat (P = I²R). If current is too high or ventilation is poor, components can overheat, melt insulation, or ignite nearby materials. Always check current ratings!
Battery Hazards
Reverse polarity: Connecting a battery backward can destroy polarized components (electrolytic capacitors, LEDs) — and cause explosions!
Overcharging: Can cause lithium batteries to swell, leak, or catch fire (thermal runaway)
Short circuit: A shorted battery can heat dangerously fast — never put loose batteries in a bag with metal objects
Chemical leakage: Old alkaline batteries can leak corrosive electrolyte
Common Science Olympiad hazard question: "What protects a circuit from overcurrent?" → Fuse or circuit breaker. "What component should never be reverse-connected?" → Polarized capacitor, LED, diode.
Safety Rules for DC Circuits
Always disconnect power before modifying a circuit
Use properly rated fuses/breakers for expected currents
Check polarity before connecting polarized components
Never exceed voltage/current ratings of components
Keep flammable materials away from circuits
Use anti-static precautions near sensitive electronics
14. Real-World DC Applications
Application
DC Component
Why DC?
Smartphones/tablets
Li-ion battery, 3.7V typical
Portable, rechargeable, consistent voltage
Laptops
Li-ion battery, 11–20V
Needs stable DC; AC adapter converts from outlet
Electric vehicles (EVs)
Large Li-ion pack (300–800V DC)
Motors can run on DC; fast charging = DC
LED lighting
LED strips, 12V DC
LEDs require DC; efficient, long-lasting
Solar panels
Photovoltaic cells, 12–48V
Cells produce DC directly from sunlight
Flashlights
AA/AAA batteries, 1.5V×cells
Simple, portable DC source
Remote controls
AAA or coin cell, 1.5–3V
Low-power electronics need stable DC
Computer motherboards
ATX supply: 3.3V, 5V, 12V DC
CPUs and RAM require precise DC voltages
DC in Modern Technology
🔋 The Battery Revolution
Li-ion batteries transformed portable electronics. Laptops, phones, EVs all depend on DC power. Early Tesla Model S packs used ~7,000 small cylindrical cells (18650 format); newer EVs use fewer, larger cells (e.g., 4680 format).
☀️ Solar Energy
Solar panels produce DC. Inverters convert DC → AC for home use, or batteries store DC directly. The world is gradually shifting toward DC microgrids for efficiency.
💡 Fun Fact
When you plug in your phone charger, it converts 120V AC (from the outlet) to ~5V DC. The "brick" is called a switched-mode power supply (SMPS). Without it, your phone would explode in milliseconds!
15. Key Person: Alessandro Volta
Historical Figure
Alessandro Volta (1745–1827) was an Italian physicist who invented the first true battery — the voltaic pile — in 1799 (published 1800).
Nationality: Italian
Era: 18th–19th century
Famous for: Inventing the first electrochemical battery (voltaic pile, 1799)
The voltaic pile: Alternating zinc and copper discs separated by saltwater-soaked cloth — stacked to produce a sustained electrical current
Unit named after him: The volt (V) — the SI unit of electric potential difference
Influenced by: Luigi Galvani's experiments with frog legs (animal electricity)
Impact: First device to produce a steady, continuous electrical current — launched the age of electrical science
Science Olympiad loves: "Who invented the battery?" → Volta. "What is voltage named after?" → Volta. Don't confuse with Ampere (current) or Ohm (resistance)!
Remember: Volt → Volta → Voltage. The unit is named after the person who made sustained voltage possible!
16. Practice Problems (15 Questions)
Select an answer or type your response, then click Submit to check yourself.
Q1
Three resistors of 10Ω, 15Ω, and 25Ω are connected in series to a 25V battery. What is the total resistance and current?
Q2
Two resistors of 12Ω and 6Ω are connected in parallel. What is their combined resistance?
Q3
A battery has an EMF of 6V and internal resistance of 0.2Ω. If 3A of current flows, what is the terminal voltage delivered to the circuit?
Q4
Which statement about parallel circuits is correct?
Q5
A 40Ω resistor carries 0.3A of current. What is the voltage across it, and how much power does it dissipate?
Q6
Two identical 60W bulbs are connected in parallel across 120V. What is the total current drawn from the source?
Q7
In a series circuit with a 24V battery, three equal resistors each have a voltage of 8V across them. What is the resistance of each if the current is 0.4A?
Q8
What type of energy does a battery store, and how does it produce electrical energy?
Q9
Three 30Ω resistors are connected in parallel. What is the combined resistance?
Q10
You have a circuit with a 9V battery, a 27Ω resistor, and an LED in series. The LED has a forward voltage of 2V. What current flows through the LED, and what is its power consumption?
Q11
A 100W light bulb runs for 8 hours every day for 30 days. How many kilowatt-hours of energy does it use?
Q12
What happens to the other bulbs in a series string when one bulb burns out? Why? What about in a parallel circuit?
Q13
Which scientist invented the first battery, and what is the unit of voltage named after them?
Q14
A fuse rated at 5A is placed in a circuit. The circuit has a 12V battery and three 2Ω resistors in series. Will the fuse blow?
Q15
In a parallel circuit, resistors of 4Ω, 6Ω, and 12Ω are connected to a 12V battery. Find: (a) total resistance, (b) total current, (c) current through each branch.
17. 📝 Cheat Sheet — Topic 2: DC Circuits
Concept
Key Facts
DC (Direct Current)
One direction only. Constant voltage. 0 Hz. From batteries.
Battery
Chemical → electrical energy. Anode (−) oxidizes, cathode (+) reduces. Cell vs. battery.
Series Batteries
Vtotal = V₁+V₂+... (voltages add)
Parallel Batteries
Voltage unchanged, capacity (life) adds
Internal Resistance
Vterminal = EMF − Ir
Ohm's Law
V = IR | I = V/R | R = V/I
Series Circuits
Rtotal=ΣR. Same I. V divides. One fails → all fail.
Parallel Circuits
1/Rtotal=Σ(1/R). Same V. I divides. One fails → others work.
Power
P = IV = I²R = V²/R (Watts)
Energy
E = Pt (Joules) or kWh
Fuse
Protects against overcurrent. Always in series. One-time use.
LED
Light from current (one direction). Needs current-limiting resistor.
Volta (person)
Italian physicist (1745–1827). Invented voltaic pile (first battery). Volt named after him.
Formula Quick Reference — DC Circuits:
V = IR | Rseries = R₁+R₂+... | 1/Rparallel = 1/R₁+1/R₂+...
P = IV = I²R = V²/R | Vterminal = EMF − Ir
Series: same I, V divides | Parallel: same V, I divides
📺 Video References — DC Circuits & Batteries
Curated Khan Academy and YouTube videos for middle school Science Olympiad prep. Watch alongside the study guide for deeper understanding.
Why some fairy lights go dark when one bulb fails but home lights keep shining. Perfect middle school explanation of series vs parallel with real-world examples.
📘 Made for middle school — start here for Sections 9–11
Build your own circuits with batteries, resistors, switches, and light bulbs. Test series vs parallel, measure voltage and current. Free from University of Colorado.
🎮 Interactive — the absolute best way to learn circuits by doing
💡 Study tip: Start with the 5–6 min middle school videos, then watch the Khan Academy Physics series for competition depth. Use the PhET Circuit Construction Kit to build circuits yourself — it's the single best practice tool!
1. What Is Alternating Current (AC)?
Key Definition
Alternating Current (AC) is electric current that periodically reverses direction. Unlike DC (which flows one way), AC electrons oscillate back and forth — like a piston, not a river.
In AC, the voltage swings between positive and negative values following a sine wave pattern. The current direction flips many times per second.
DC vs. AC at a Glance
DC is a flat line (constant). AC is a sine wave (oscillating).
"AC = Alternating = Always Changing direction." The current literally alternates which way it flows. 60 times per second in the US!
If the test asks "What comes out of your wall outlet?" → AC, 120V RMS, 60 Hz. That's the trio to memorize.
2. AC Waveforms — Anatomy of a Sine Wave
Key Concept
The most common AC waveform is the sine wave. Every AC quantity — voltage, current — follows this smooth, repeating pattern described by: v(t) = Vpeak sin(2πft)
One complete cycle: positive peak → zero → negative peak → zero. The period T is the time for one full cycle.
Key Terms
Term
Symbol
Meaning
Amplitude
Vpeak
Maximum value from zero (the height of the wave)
Peak-to-Peak
Vpp
Distance from positive peak to negative peak = 2 × Vpeak
Period
T
Time for one complete cycle (seconds)
Frequency
f
Number of cycles per second (Hertz)
Wavelength
λ
Physical length of one cycle (for electromagnetic waves)
v(t) = Vpeak · sin(2πft)
v(t) = instantaneous voltage at time tVpeak = amplitude (maximum voltage)f = frequency in Hzt = time in seconds
The amplitude is measured from the center line to ONE peak, NOT from peak to peak. Vpp = 2 × Vpeak.
3. AC Voltage & Current — Peak vs. RMS
Key Concept
Since AC voltage constantly changes, we need a way to express its "effective" value. RMS (Root Mean Square) is the AC value that delivers the same power as an equivalent DC value.
VRMS = Vpeak / √2 ≈ 0.707 × Vpeak
Similarly: IRMS = Ipeak / √2RMS = "Root Mean Square" — square all values, take the mean, then the square rootWhen someone says "120V AC," they mean 120V RMS!
💡 Real World
US wall outlet = 120V RMS
Vpeak = 120 × √2 = 170V
That means the voltage actually swings from +170V to −170V — but the effective (RMS) value is 120V. That's the number on the label because it matches the DC equivalent for power delivery.
Why RMS Matters
A 120V RMS AC source delivers the same heating power to a resistor as a 120V DC source. RMS is the "fair comparison" between AC and DC.
"RMS ≈ 70.7% of Peak" — Quick shortcut: multiply peak by 0.707 to get RMS, or multiply RMS by 1.414 to get peak.
4. Frequency & Period
Key Relationship
Frequency (f) is how many complete cycles happen per second. Period (T) is the time for one cycle. They are reciprocals: f = 1/T.
f = 1/T | T = 1/f | ω = 2πf
f = frequency in Hertz (Hz) = cycles/secondT = period in secondsω = angular frequency in radians/second
Region
Frequency
Period
🇺🇸 United States
60 Hz
16.67 ms
🇪🇺 Europe / 🇮🇳 India
50 Hz
20 ms
Aircraft electrical
400 Hz
2.5 ms
Worked Example
Q: US wall power is 60 Hz. Find the period and angular frequency.
T = 1/f = 1/60 = 0.01667 s = 16.67 ms
ω = 2πf = 2π(60) = 377 rad/s
The test may ask: "What frequency is US household power?" → 60 Hz. "Period?" → 1/60 s ≈ 16.7 ms. Know both!
5. AC Power Generation — Generators
Key Concept
An AC generator (alternator) converts mechanical energy into electrical energy by rotating a coil inside a magnetic field. This is Faraday's law in action: a changing magnetic flux through a coil induces an EMF.
A coil spins in a magnetic field. As the flux through the coil changes, an alternating EMF is induced. Slip rings maintain connection to the spinning coil.
EMF = NBAω sin(ωt)
N = number of turns in the coilB = magnetic field strength (Tesla)A = area of the coil (m²)ω = angular velocity (rad/s)
Why it produces a sine wave: As the coil rotates, the angle between the coil and the magnetic field continuously changes. The flux through the coil is Φ = BA cos(θ), and the induced EMF is the rate of change of flux, which gives a sine function.
"Spin to Win" — Power plants spin turbines (using steam from coal/nuclear/gas, or water/wind) to rotate generators. The spinning IS the energy conversion.
A generator uses slip rings (AC output). A DC motor/generator uses a commutator (splits the ring to flip connections each half-turn). Don't mix them up!
6. Transformers
Key Concept
A transformer changes AC voltage levels using electromagnetic induction between two coils. It only works with AC — because a changing current is needed to create a changing magnetic flux.
Step-down transformer: more primary turns, fewer secondary turns → lower output voltage. The iron core channels magnetic flux between coils.
V₁/V₂ = N₁/N₂ = I₂/I₁
V₁, V₂ = primary, secondary voltageN₁, N₂ = number of turns on each coilI₁, I₂ = primary, secondary currentPower is conserved (ideal): P₁ = P₂ → V₁I₁ = V₂I₂
Step-Up Transformer
N₂ > N₁ → V₂ > V₁ Increases voltage, decreases current. Used: Power plant → transmission lines (increases to 500kV+ for efficient long-distance transport).
Step-Down Transformer
N₂ < N₁ → V₂ < V₁ Decreases voltage, increases current. Used: Transmission lines → your house (reduces from thousands of volts to 120/240V).
Worked Example
Q: A transformer has 500 primary turns and 100 secondary turns. If V₁ = 240V, find V₂ and the turn ratio.
Transformers only work with AC! DC creates a constant magnetic field — no changing flux, no induced EMF. This is the fundamental reason AC won the "War of Currents."
The test loves asking "Why can't transformers work with DC?" → Because DC doesn't change, so it can't create a changing magnetic flux, which is required for electromagnetic induction.
7. Impedance & Reactance
Key Concept
In AC circuits, opposition to current comes from three sources: Resistance (R) from resistors, Capacitive Reactance (XC) from capacitors, and Inductive Reactance (XL) from inductors. The total opposition is called Impedance (Z).
Reactance
Capacitive Reactance (XC)
XC = 1 / (2πfC)
Measured in Ohms (Ω). Decreases with higher frequency — capacitors "pass" high frequencies and "block" low frequencies/DC.
Inductive Reactance (XL)
XL = 2πfL
Measured in Ohms (Ω). Increases with higher frequency — inductors "block" high frequencies and "pass" low frequencies/DC.
Z = √(R² + (XL − XC)²)
Z = impedance (Ω) — the AC equivalent of resistanceR = resistance (Ω)XL = 2πfL (inductive reactance)XC = 1/(2πfC) (capacitive reactance)
Worked Example
Q: An AC circuit at 60 Hz has R = 30Ω, L = 0.1H, and C = 100μF. Find XL, XC, and Z.
"ELI the ICE man" — In an inductor (L), voltage (E) leads current (I). In a capacitor (C), current (I) leads voltage (E). This is the #1 mnemonic for AC circuits!
Reactance is NOT resistance. Resistance dissipates energy as heat. Reactance stores and returns energy — capacitors in electric fields, inductors in magnetic fields. Only resistance appears in power loss.
8. RLC Circuits & Resonance
Key Concept
A series RLC circuit contains a Resistor, Inductor, and Capacitor in series. At a special frequency called the resonant frequency, XL = XC, they cancel, and impedance drops to its minimum (just R).
f₀ = 1 / (2π√(LC))
f₀ = resonant frequency (Hz)L = inductance (Henrys)C = capacitance (Farads)At resonance: XL = XC, so Z = R (minimum impedance, maximum current)
Phasor Diagram
VR is in phase with current. VL leads by 90°. VC lags by 90°. Total voltage is the vector sum.
Worked Example
Q: Find the resonant frequency of a circuit with L = 10mH and C = 100μF.
"At resonance, Z = R." This means maximum current flows. Radio tuners exploit this — tune L or C until f₀ matches the station frequency, maximizing that signal.
9. AC Power — Real, Reactive & Apparent
Key Concept
In AC circuits, current and voltage may be out of phase. This means not all the power delivered actually does useful work. We define three types of power.
Type
Symbol
Unit
Formula
Meaning
Real Power
P
Watts (W)
P = VI cos(φ)
Actual work done (heat, light, motion)
Reactive Power
Q
VAR
Q = VI sin(φ)
Energy stored/returned by L and C
Apparent Power
S
VA
S = VI
Total power delivered by source
Power Factor = cos(φ) = P/S = R/Z
φ = phase angle between voltage and currentcos(φ) = 1 → purely resistive (all power is real)cos(φ) = 0 → purely reactive (no real power delivered)
S² = P² + Q². Real power P does useful work. Reactive power Q oscillates between source and L/C.
Worked Example
Q: An AC circuit draws 5A RMS at 120V RMS with a phase angle of 30°. Find P, Q, and S.
S = VI = 120 × 5 = 600 VA
P = VI cos(30°) = 600 × 0.866 = 519.6 W
Q = VI sin(30°) = 600 × 0.5 = 300 VAR
Power factor = cos(30°) = 0.866
"Power factor = how efficient your AC circuit is." A PF of 1.0 means all power does work. Industrial facilities pay penalties for low power factor!
10. RC & RL Circuits — Time Constants
Key Concept
When a capacitor charges/discharges through a resistor (RC circuit), or current builds up/decays through an inductor (RL circuit), the process is not instant — it follows an exponential curve governed by the time constant τ.
RC Circuit
τ = RC
τ = time constant (seconds)
R = resistance (Ω), C = capacitance (F)
After 1τ: capacitor reaches 63.2% of final value
After 5τ: effectively fully charged (99.3%)
RL Circuit
τ = L/R
τ = time constant (seconds)
L = inductance (H), R = resistance (Ω)
After 1τ: current reaches 63.2% of final value
After 5τ: effectively at full current (99.3%)
"5τ and you're done." After 5 time constants, capacitors are 99.3% charged — close enough to call it complete. Most tests accept 5τ as "fully charged."
11. AC Applications
Key Concept
AC won the "War of Currents" because it can be easily transformed to high voltage for efficient long-distance transmission, then stepped down for safe home use.
The Power Grid
Stage
Voltage
Purpose
Power plant generator
~11,000–25,000V
Generate AC power
Step-up transformer
115,000–765,000V
High voltage for efficient long-distance transmission
Transmission lines
115,000–765,000V
Low current → less I²R loss in wires
Step-down substation
~7,200–13,800V
Distribution to neighborhoods
Pole transformer
120/240V
Safe voltage for your home
Why High Voltage for Transmission?
Key Insight
Power loss in wires = Ploss = I²R. If you increase voltage by 10×, current drops by 10× (since P = VI stays the same). Power loss drops by 10² = 100×! This is why we transmit at hundreds of thousands of volts.
Common AC Applications
🏠 Household
Wall outlets, lighting, kitchen appliances, HVAC, washing machines — all run on AC (120V/60Hz in the US).
⚡ Motors
AC induction motors run fans, pumps, compressors, and industrial machinery. AC's alternating field naturally creates rotation — no commutator needed.
🎵 Audio
Sound signals are AC — speakers use alternating current to vibrate a cone back and forth, creating sound waves.
📻 Radio
Radio signals are high-frequency AC (kHz to GHz). LC resonance circuits tune to specific frequencies to pick up stations.
12. Key Person: Nikola Tesla (1856–1943)
Key Figure
Nikola Tesla was a Serbian-American inventor and engineer who developed the modern AC power system, including the AC induction motor and polyphase power distribution. He won the "War of Currents" against Edison's DC system.
Major Contributions
AC induction motor — simple, reliable, no brushes/commutators. Still used everywhere today.
Polyphase AC system — three-phase power for efficient generation and distribution.
Tesla coil — resonant transformer producing spectacular high-voltage, high-frequency sparks.
Niagara Falls power plant (1895) — first large-scale AC hydroelectric plant, proved AC could power a city.
Held ~300 patents across 26 countries.
The War of Currents
In the 1880s–90s, Thomas Edison championed DC while Tesla (backed by George Westinghouse) promoted AC. Edison ran a smear campaign — publicly electrocuting animals to show AC was "dangerous." But AC's ability to transform voltage (transmit over long distances cheaply) ultimately won. The 1893 World's Fair in Chicago, powered by Westinghouse's AC system, sealed the victory.
Test fact: The SI unit of magnetic flux density is the Tesla (T), named after Nikola Tesla.
"Tesla = AC, Edison = DC." Tesla championed the system that actually powers the world. The unit Tesla (T) measures magnetic field strength.
13. Practice Problems (15 Questions)
Question 1 (Multiple Choice)
What is the frequency of AC power in the United States?
Question 2 (Free Response)
A US wall outlet provides 120V RMS. Calculate the peak voltage and peak-to-peak voltage.
Question 3 (Multiple Choice)
Which mnemonic helps remember the phase relationship between voltage and current in inductors and capacitors?
Question 4 (Free Response)
A transformer has 200 primary turns and 50 secondary turns. If the input voltage is 480V, what is the output voltage? Is this step-up or step-down?
Question 5 (Multiple Choice)
Why don't transformers work with DC?
Question 6 (Free Response)
Calculate the capacitive reactance of a 47μF capacitor at 60 Hz. Then calculate it at 1000 Hz. What happens to X_C as frequency increases?
Question 7 (Free Response)
An AC circuit has R = 40Ω, X_L = 50Ω, and X_C = 20Ω. Calculate the impedance Z.
Question 8 (Multiple Choice)
At resonance in a series RLC circuit, what is the impedance equal to?
Question 9 (Free Response)
Find the resonant frequency of a circuit with L = 5mH and C = 200μF.
Question 10 (Free Response)
An AC circuit draws 10A RMS at 240V RMS with a power factor of 0.8. Calculate the real power (P), apparent power (S), and reactive power (Q).
Question 11 (Multiple Choice)
What does RMS stand for, and why is it used for AC measurements?
Question 12 (Free Response)
A 22kΩ resistor is in series with a 10μF capacitor. Calculate the time constant τ. How long until the capacitor is approximately fully charged?
Question 13 (Multiple Choice)
Who developed the modern AC power system and won the "War of Currents"?
Question 14 (Free Response)
A step-up transformer takes 120V input and outputs 600V. If the primary coil has 100 turns, how many turns does the secondary coil have? If the primary current is 10A, what is the secondary current (ideal transformer)?
Question 15 (Free Response)
Explain why power companies transmit electricity at very high voltages (hundreds of thousands of volts) rather than at 120V. Use the formula P_loss = I²R in your explanation.
14. Cheat Sheet — AC Circuits
Concept
Formula / Fact
AC waveform
v(t) = Vpeak sin(2πft)
RMS voltage
VRMS = Vpeak / √2 ≈ 0.707 × Vpeak
RMS current
IRMS = Ipeak / √2
US wall power
120V RMS, 60 Hz
Period ↔ Frequency
f = 1/T, T = 1/f
Angular frequency
ω = 2πf
Generator EMF
EMF = NBAω sin(ωt)
Transformer ratio
V₁/V₂ = N₁/N₂ = I₂/I₁
Capacitive reactance
XC = 1/(2πfC)
Inductive reactance
XL = 2πfL
Impedance
Z = √(R² + (XL − XC)²)
Resonant frequency
f₀ = 1/(2π√(LC))
At resonance
XL = XC, Z = R, max current
Real power
P = VI cos(φ) [Watts]
Reactive power
Q = VI sin(φ) [VAR]
Apparent power
S = VI [VA]
Power factor
cos(φ) = P/S = R/Z
RC time constant
τ = RC
RL time constant
τ = L/R
Fully charged/discharged
≈ 5τ (99.3%)
Phase mnemonic
ELI the ICE man
Power loss in wires
Ploss = I²R (high V → low I → less loss)
Tesla unit
T = magnetic flux density, named after Nikola Tesla
Time constants, exponential curves, RC charging/discharging
Intermediate
1. Voltage, Current & Resistance — The Big Three
Key Concept
These three quantities define every circuit. Voltage pushes, Current flows, Resistance opposes. They're connected by Ohm's Law: V = IR.
The Water Analogy
Voltage = water pressure. Current = flow rate. Resistance = pipe narrowing. Higher pressure + wider pipe = more flow!
Quantity
Symbol
Unit
Measures
Water Analogy
Voltage
V
Volts (V)
Electric potential difference (energy per charge)
Water pressure
Current
I
Amperes (A)
Rate of charge flow (coulombs/second)
Flow rate (liters/sec)
Resistance
R
Ohms (Ω)
Opposition to current flow
Pipe narrowing
V = IR | I = V/R | R = V/I
Ohm's Law — the most important equation in circuit analysisV = voltage in Volts, I = current in Amps, R = resistance in Ohms
Important Definitions
Voltage (V) = the energy per unit charge. 1 Volt = 1 Joule per Coulomb. It's the "push" that makes electrons move.
Current (I) = the rate of charge flow. 1 Ampere = 1 Coulomb per second. It's how many electrons pass a point each second.
Resistance (R) = opposition to current. 1 Ohm = resistance when 1V produces 1A. Materials with loosely bound electrons have low R (conductors).
"VIR" triangle: Cover the one you want to find. V on top, I and R on the bottom. V = IR, I = V/R, R = V/I.
2. Kirchhoff's Voltage Law (KVL)
Key Law
The sum of all voltages around any closed loop in a circuit equals zero.
∑V = 0 (around any closed loop)
In other words: what the battery puts in, the components use up — perfectly balanced.
Following the loop clockwise: battery provides +12V, resistors drop −4V, −3V, −5V. Sum = 0. Energy is conserved!
Worked Example
Q: In a series circuit with a 24V battery and three resistors, V₁ = 8V and V₂ = 10V. What is V₃?
"What goes up must come down." The voltage the battery provides MUST be completely used up by the components. No voltage is lost or created — it's conservation of energy applied to circuits.
KVL is your go-to tool for finding an unknown voltage drop. Set up the loop equation, plug in what you know, and solve for the unknown.
3. Kirchhoff's Current Law (KCL)
Key Law
The sum of all currents entering a junction equals the sum of all currents leaving.
∑Iin = ∑Iout (at any node/junction)
Charge cannot pile up or disappear at a junction — what comes in must go out.
5A flows in, 5A flows out. Current is conserved at every junction — like cars at a highway interchange.
Worked Example
Q: At a junction, three wires carry: 6A in, 2A out, and I₃ unknown out. Find I₃.
KCL: ∑Iin = ∑Iout
6 = 2 + I₃
I₃ = 4A (out)
KCL applies at every junction, not just simple ones. In complex circuits with many branches, pick a node, label all currents in/out, and set up the equation.
4. Voltage Divider Rule
Key Formula
A voltage divider is two (or more) resistors in series. The voltage across each resistor is proportional to its resistance relative to the total.
Vout = Vin × R₂ / (R₁ + R₂)
Vout = voltage across R₂ (the "bottom" resistor)Vin = total source voltageR₁ = "top" resistor, R₂ = "bottom" resistor
Vout is taken across R₂. The bigger R₂ is relative to R₁, the more voltage it gets.
"Bigger resistor gets bigger share." The voltage drop across each resistor is proportional to its fraction of the total resistance.
Voltage dividers appear in sensor circuits, volume controls (potentiometers), and test questions. Know this formula cold.
5. Current Divider Rule
Key Formula
When two resistors are in parallel, the total current splits between them. The smaller resistor gets the larger share of current (path of least resistance).
I₁ = Itotal × R₂ / (R₁ + R₂)
I₁ = current through R₁Notice: the "other" resistor (R₂) is in the numerator — NOT the same resistor!For N resistors: Ik = Itotal × (Rtotal/Rk)
Worked Example
Q: R₁ = 4Ω and R₂ = 12Ω are in parallel. Total current is 8A. Find I₁ and I₂.
Current divider uses the OTHER resistor in the numerator! This is opposite from the voltage divider (which uses the same resistor). It's a classic trap.
6. Thévenin's Theorem
Key Theorem
Any linear circuit with two terminals can be replaced by a single voltage source VTh in series with a single resistor RTh. This simplifies complex circuits into something trivial to analyze.
Any complex linear circuit seen from terminals A-B behaves identically to VTh + RTh.
How to Find Thévenin Equivalent
Remove the load (disconnect whatever's between terminals A and B)
Find VTh = Open-circuit voltage between A and B (voltage with nothing connected)
Find RTh = "Turn off" all independent sources (replace voltage sources with short circuits, current sources with open circuits) and find the resistance looking into A-B
Worked Example
Q: A circuit has a 12V battery in series with R₁ = 4Ω, then splits to R₂ = 6Ω in parallel with terminals A-B. Find VTh and RTh.
VTh: With A-B open, no current flows through R₂ (it's in a dead branch). All 12V drops across R₁? No — with A-B open, no current flows at all through R₁ either (it's an open circuit). VTh = 12V (full battery voltage appears at A-B).
Wait — if R₂ is in parallel with A-B: VTh = V across R₂ = 12 × R₂/(R₁+R₂) = 12 × 6/(4+6) = 7.2V RTh: Short the 12V source. RTh = R₁ ∥ R₂ = (4×6)/(4+6) = 24/10 = 2.4Ω
Thévenin's theorem is extremely useful for finding the current/voltage at a specific load in a complex circuit. Reduce everything else to VTh + RTh, then use Ohm's law!
7. Norton's Theorem
Key Theorem
Norton's theorem is the "twin" of Thévenin's. Any linear circuit can be replaced by a single current source IN in parallel with a resistor RN.
IN = VTh / RTh | RN = RTh
IN = Norton (short-circuit) currentRN = Norton resistance = same as Thévenin resistance!Thévenin ↔ Norton are interconvertible
How to Find Norton Equivalent
Short-circuit A-B and find the current flowing through the short → that's IN
RN = RTh (found the same way — deactivate sources, find resistance)
Worked Example
Q: From the previous Thévenin example (VTh = 7.2V, RTh = 2.4Ω), find the Norton equivalent.
IN = VTh/RTh = 7.2/2.4 = 3A
RN = RTh = 2.4Ω
Norton equivalent: 3A current source in parallel with 2.4Ω
"Thévenin = voltage source + series R. Norton = current source + parallel R." Same R, just different representation. Convert between them anytime!
8. Superposition Theorem
Key Theorem
In a linear circuit with multiple sources, the response (voltage/current) at any point equals the algebraic sum of responses caused by each source acting alone, with all other sources deactivated.
Steps
Activate one source, deactivate all others (voltage sources → short circuit, current sources → open circuit)
Find the response (voltage or current at the point of interest)
Repeat for each source
Add all responses algebraically (mind the signs!)
Worked Example
Q: A circuit has V₁ = 10V and V₂ = 6V, both connected to a 5Ω resistor (V₁ through 2Ω, V₂ through 3Ω). Find the current through the 5Ω resistor.
Audio speakers use impedance matching (8Ω speaker with 8Ω amplifier output) for maximum power transfer. This is why speaker impedance matters!
10. Measurement Instruments
Key Rules
Ammeter: measures current. Connected in series. Must have very low internal resistance (ideally 0Ω) so it doesn't affect the circuit.
Voltmeter: measures voltage. Connected in parallel. Must have very high internal resistance (ideally ∞Ω) so it draws negligible current.
🔌 Ammeter (Series)
Break the circuit, insert ammeter in the gap. Current flows through it. Low R so it doesn't reduce current. Danger: Connecting an ammeter in parallel → near-short-circuit → blown fuse!
⚡ Voltmeter (Parallel)
Connect across the component you want to measure. Measures potential difference. High R so it draws almost no current. Danger: A low-R voltmeter would draw significant current and give wrong readings.
Multimeter & Oscilloscope
Instrument
Measures
Key Feature
Multimeter
V, I, R (and often capacitance, frequency, diode test)
All-in-one tool, digital display, portable
Oscilloscope
Voltage vs. time waveform
Shows wave shape, frequency, amplitude, phase — essential for AC analysis
NEVER connect an ammeter in parallel! Its near-zero resistance will create a short circuit. Ammeters go in SERIES only. This destroys instruments and blows fuses.
"A for Amps, A for 'Along the wire' (series). V for Volts, V for 'a-V-cross' (parallel)."
11. Wheatstone Bridge
Key Concept
A Wheatstone bridge is a circuit with four resistors in a diamond arrangement and a galvanometer (sensitive current meter) across the middle. When the bridge is balanced, no current flows through the galvanometer.
When R₁/R₂ = R₃/R₄, the bridge is balanced and no current flows through the galvanometer G.
Balanced when: R₁/R₂ = R₃/R₄
Or equivalently: R₁ × R₄ = R₂ × R₃ (cross-multiplication)When balanced: VB = VC, so no current through galvanometer
Worked Example
Q: R₁ = 100Ω, R₂ = 200Ω, R₃ = 150Ω. What value of R₄ balances the bridge?
Precision resistance measurement — used to measure unknown resistances with high accuracy
Strain gauges — tiny resistance changes in a wire under stress are detected by an unbalanced Wheatstone bridge
Temperature sensors — thermistors in a bridge circuit detect small temperature changes
The Wheatstone bridge is a favorite Science Olympiad topic. Practice setting up the balance equation: R₁/R₂ = R₃/R₄. Know that it's used for precision measurements.
12. Key Person: Gustav Kirchhoff (1824–1887)
Key Figure
Gustav Robert Kirchhoff was a German physicist who formulated the two fundamental laws of circuit analysis — Kirchhoff's Current Law (KCL) and Kirchhoff's Voltage Law (KVL) — in 1845, when he was just 21 years old!
Major Contributions
Kirchhoff's Laws (1845) — KCL and KVL, the foundation of all circuit analysis
Spectroscopy — co-discovered that each chemical element has a unique spectral signature (with Robert Bunsen)
Kirchhoff's Law of Thermal Radiation — relationship between emission and absorption of radiation
Discovered the elements cesium and rubidium through spectral analysis
"Kirchhoff's laws are conservation laws in disguise." KVL = conservation of energy. KCL = conservation of charge. Physics always conserves!
13. Practice Problems (15 Questions)
Question 1 (Free Response)
A series circuit has a 15V battery and three resistors with voltage drops of 5V, 3V, and V₃. Use KVL to find V₃.
Question 2 (Multiple Choice)
Kirchhoff's Current Law states that at any junction:
Question 3 (Free Response)
At a junction, currents of 4A and 6A flow in. Currents of 3A and I₃ flow out. Find I₃ using KCL.
Question 4 (Free Response)
A voltage divider has R₁ = 3kΩ and R₂ = 7kΩ with V_in = 20V. Find V_out across R₂.
Question 5 (Multiple Choice)
How should an ammeter be connected in a circuit?
Question 6 (Free Response)
Two resistors R₁ = 10Ω and R₂ = 40Ω are in parallel with 5A total current. Use the current divider rule to find the current through each resistor.
Question 7 (Free Response)
A Thévenin equivalent circuit has V_Th = 30V and R_Th = 10Ω. If a 20Ω load is connected, find the current through and voltage across the load.
Question 8 (Multiple Choice)
In the Wheatstone bridge balanced condition, which equation is correct?
Question 9 (Free Response)
Convert a Thévenin equivalent (V_Th = 24V, R_Th = 6Ω) to its Norton equivalent.
Question 10 (Multiple Choice)
For maximum power transfer, the load resistance should be:
Question 11 (Free Response)
A Wheatstone bridge has R₁ = 120Ω, R₂ = 480Ω, R₃ = 200Ω. Find R₄ for a balanced bridge.
Question 12 (Free Response)
A Thévenin circuit has V_Th = 50V and R_Th = 25Ω. What is the maximum power that can be delivered to a load? What load resistance achieves this?
Question 13 (Multiple Choice)
When applying superposition, how do you "deactivate" a voltage source?
Question 14 (Free Response)
Explain why a voltmeter must have very high internal resistance and an ammeter must have very low internal resistance. What would happen if these were reversed?
Question 15 (Free Response)
A circuit has a 36V battery with two loops. Loop 1 has R₁ = 6Ω and R₂ = 12Ω in series. Loop 2 shares R₂ and has R₃ = 4Ω. Using KVL for both loops, find the current through R₂. (Hint: define loop currents I₁ and I₂.)
14. Cheat Sheet — Circuit Fundamentals
Concept
Formula / Fact
Ohm's Law
V = IR, I = V/R, R = V/I
Power
P = IV = I²R = V²/R
KVL (Voltage Law)
∑V around a loop = 0 (conservation of energy)
KCL (Current Law)
∑I at a node = 0, or Iin = Iout (conservation of charge)
Voltage divider
Vout = Vin × R₂/(R₁ + R₂)
Current divider
I₁ = Itotal × R₂/(R₁ + R₂) — uses the OTHER resistor!
Thévenin equivalent
VTh (open-circuit V) + RTh (deactivated-source R) in series
Norton equivalent
IN = VTh/RTh, RN = RTh, in parallel
Superposition
One source at a time; V-source → short, I-source → open
Max power transfer
Rload = RTh, Pmax = VTh²/(4RTh)
Ammeter
Series connection, very low R
Voltmeter
Parallel connection, very high R
Wheatstone bridge
Balanced: R₁/R₂ = R₃/R₄ (or R₁R₄ = R₂R₃)
Kirchhoff's Laws
Named after Gustav Kirchhoff (1824–1887), formulated 1845