WattsUp

Basics

Circuits

Power & V

Complex

Statics

Magnets

Lesson 01 — Basics

What is a Circuit?

A circuit is a closed loop that allows electric current to flow. It needs a power source, a conductive path, and a load to do useful work.

🔋Power SourceProvides EMF (e.g. battery)

〰️ConductorCarries current (wire)

💡LoadConverts energy (bulb, etc.)

🔄Closed LoopCurrent must have a complete path

Lesson 02 — Ohm's Law

Ohm's Law

The relationship between voltage, current, and resistance — the foundation of all circuit analysis.

V = I × R

I = V / RR = V / I

Voltage (V)

9 V

Resistance (Ω)

100 Ω

0.09
Current (A)

90
Current (mA)

0.81
Power (W)

Lesson 03 — Series

Series Circuits

In a series circuit, components are connected end-to-end in a single path. The same current flows through every component.

Same ICurrent is identical through every component

V addsV₁ + V₂ + V₃ = V_supply

R addsR_total = R₁ + R₂ + R₃

Lesson 04 — Parallel

Parallel Circuits

In a parallel circuit, components share the same two nodes. Each branch has the same voltage but can carry different currents.

Same VVoltage is identical across every branch

I addsI_total = I₁ + I₂ + I₃

1/R1/R_total = 1/R₁ + 1/R₂ + 1/R₃

Lesson 05 — Power & Energy

Electrical Power & Energy

Power is the rate of energy transfer (watts). Energy is power sustained over time — what you actually pay for on your electricity bill (kilowatt-hours).

P = V × IVoltage × Current

P = I²RCurrent² × Resistance

E = P × tPower × Time (J or kWh)

Voltage (V)

12 V

Current (A)

1.00 A

Time (hours)

1.0 h

12.00
Power (W)

43200
Energy (J)

0.0120
Energy (kWh)

0 W2400 W max

💡LED Bulb — 10 WRuns 100 h on 1 kWh

🖥️Laptop — 65 W1 kWh lasts ~15 hours

🔌Kettle — 2000 WBoils in ~2 min, uses 0.07 kWh

📱Phone charger — 5 WFull charge ≈ 0.015 kWh

Lesson 06 — Voltage Divider

Voltage Divider

Two resistors in series split the supply voltage proportionally. Vout tapped between them scales with the ratio of R2 to the total resistance.

FormulaVout = Vin × R2 / (R1 + R2)

Input Voltage Vin

12 V

R1 — top (Ω)

1000 Ω

R2 — bottom (Ω)

1000 Ω

6.00
Vout (V)

50.0
Ratio (%)

6.00
Current (mA)

SensorsThermistors & LDRs shift Vout as temperature or light changes R2

ADC inputScale a 5V signal down to 3.3V for a microcontroller ADC pin

BiasSet a midpoint reference voltage in amplifier circuits

Lesson 07 — Electric Potential

Electric Potential

Electric potential is the electric potential energy stored per unit charge at a point. Voltage is the difference in potential between two points — it drives current to flow.

V = W / QWork per unit charge (J/C)

ΔV = Vₐ − VₙPotential difference

W = Q × VWork done moving charge Q

Charge Q (mC)

100 mC

Potential Difference ΔV

12 V

1200.000
Work W (mJ)

1.20000
Work W (J)

⚡Electric PotentialEnergy per unit charge at a point — measured in Volts (J/C)

🔋EMFElectromotive force — potential difference supplied by the source

〰️EquipotentialLines of equal potential — no work needed to move charge along them

📐Ground (0 V)Potential is always measured relative to a reference point

AnalogyLike gravitational PE — charges at higher potential have more stored energy

DirectionConventional current flows from high (+) to low (−) potential

Unit1 Volt = 1 Joule per Coulomb (J/C)

Lesson 08 — Complex Networks

Equivalent Resistance

Real circuits often combine series and parallel groups in multiple layers. The key technique is progressive reduction — always start with the innermost (most nested) series or parallel group, replace it with a single equivalent resistor, then work outward until one resistor remains.

Step 1Identify the innermost series or parallel group

Step 2Replace with R_eq using the series or parallel formula

Step 3Repeat until only one equivalent resistor remains

Lesson 08A

( R1 ∥ R2 ) + R3

A parallel pair feeds into a series resistor — the simplest mixed topology.

( R1 ∥ R2 ) — R3

R1 (Ω)

100 Ω

R2 (Ω)

100 Ω

R3 — series (Ω)

50 Ω

100
R eq (Ω)

Lesson 08B

R1 + ( R2 ∥ (R3 + R4) )

A series resistor followed by a parallel block where one branch itself contains two series resistors — 3-level reduction.

R1 — [ R2 ∥ (R3+R4) ]

R1 — series (Ω)

30 Ω

R2 — parallel top (Ω)

120 Ω

R3 — inner series (Ω)

60 Ω

R4 — inner series (Ω)

60 Ω

90
R eq (Ω)

Lesson 08C

(R1 + R2) ∥ (R3 + R4)

Two parallel branches, each containing two series resistors — reduce each branch first, then combine in parallel.

( R1+R2 ) ∥ ( R3+R4 )

R1 — branch A (Ω)

100 Ω

R2 — branch A (Ω)

100 Ω

R3 — branch B (Ω)

100 Ω

R4 — branch B (Ω)

100 Ω

100
R eq (Ω)

Lesson 09 — Electrostatic Force

Coulomb's Law

Any two charged objects exert a force on each other proportional to their charges and inversely proportional to the square of their separation.

F = kq₁q₂/r²Point charges

k = 8.99×10⁹N·m²/C²

F = qE = qV/dParallel plates

Point Charge Calculator

Charge q₁ (nC)

10 nC

Charge q₂ (nC)

10 nC

Separation r (cm)

10 cm

8.99
Force (µN)

8.99×10⁻⁶
Force (N)

Parallel Plate Force on Charge

Between ideal parallel plates the field is uniform: E = V/d, so the force on a charge q is F = qV/d.

Test charge q (nC)

50 nC

Plate voltage V (V)

100 V

Separation d (mm)

5 mm

20.0
E field (kV/m)

1.00
Force (µN)

⊕Like chargesSame sign → repel (force away from each other)

⊖Unlike chargesOpposite sign → attract (force toward each other)

📐Inverse squareDoubling distance quarters the force

⚡SuperpositionForces from multiple charges add as vectors

Lesson 10 — Electric Field

Electric Field

The electric field E at a point is the force per unit positive test charge placed there. It is a vector — it has both magnitude and direction.

E = F / qDefinition (N/C)

E = kQ / r²Point source

E = V / dParallel plates

Point Charge E Field

Source charge Q (nC)

20 nC

Distance r (cm)

10 cm

17980
E (N/C)

17.98
F on 1 nC (µN)

Parallel Plate E Field

Plate voltage V (V)

200 V

Separation d (mm)

5 mm

40.0
E (kV/m)

40000
E (V/m)

DirectionField points from + to − (away from positive charge)

UnitN/C = V/m (both are equivalent)

Uniform fieldBetween ideal parallel plates E is constant everywhere

Lesson 11 — Electric Potential

Potential from a Point Charge

The electric potential V at distance r from a point charge Q is the work per unit charge needed to bring a positive test charge from infinity to that point.

V = kQ / rPoint charge (Volts)

U = qVPE of charge q at V

E = −dV/drField is gradient of V

Source charge Q (nC)

20 nC

Distance r (cm)

10 cm

1798
Potential V (V)

1.798
PE of 1 nC (µJ)

🔢Scalar quantityV has no direction — unlike E it's just a number at each point

〰️EquipotentialsSurfaces of constant V — field lines always cross them perpendicularly

➕Positive chargeV > 0, potential falls as you move away

➖Negative chargeV < 0, potential rises (toward zero) as you move away

NoteV falls as 1/r while E falls as 1/r² — potential reaches farther

Lesson 07V = W/Q connects to circuit voltage — same concept, different geometry

Lesson 12 — Capacitance

Capacitance & Capacitors

A capacitor stores charge on two conducting plates separated by a gap. The ratio of stored charge to applied voltage is called capacitance C.

C = ε₀A / dParallel plate (F)

Q = C × VCharge stored (C)

U = ½CV²Energy stored (J)

Plate area A (cm²)

100 cm²

Separation d (mm)

1 mm

Applied voltage V (V)

12 V

885
Capacitance (pF)

10.62
Charge Q (nC)

63.72
Energy U (nJ)

📐Larger areaBigger plates → more charge fits → higher C

↔️Closer platesSmaller d → stronger field → higher C

🧪DielectricInsulating material between plates multiplies C by κ (dielectric constant)

⚡Energy storageCapacitors release energy instantly — used in camera flashes, defibrillators

ε₀Permittivity of free space = 8.854 × 10⁻¹² F/m

RC circuitsCapacitors + resistors give time-dependent behaviour — τ = RC

Lesson 13 — Magnetic Force

Lorentz Force

A magnetic field exerts a force on moving charges and current-carrying conductors. The force is always perpendicular to both the velocity (or current) and the field — it never does work on the charge.

F = qvB sinθMoving charge

F = BIL sinθCurrent wire

θ = 90° → maxPerpendicular = max force

Moving Charge in Field

Charge q (nC)

50 nC

Velocity v (m/s)

100 m/s

Field B (mT)

50 mT

Angle θ (°)

90°

250
Force (µN)

2.50×10⁻⁴
Force (N)

Force on Current-Carrying Wire

Field B (mT)

100 mT

Current I (mA)

500 mA

Wire length L (cm)

20 cm

Angle θ (°)

90°

1000
Force (µN)

1.00×10⁻³
Force (N)

🤚Right-hand ruleFingers along v (or I), curl toward B → thumb points in F direction

⊥Always perpendicularF ⊥ v and F ⊥ B — the force changes direction, never speed

🔄Circular motionA charge in a uniform B field orbits in a circle — basis of cyclotrons

⚡Motor principleF = BIL is the force that makes electric motors spin

Lesson 14 — Magnetic Field Sources

Creating Magnetic Fields

Moving charges and currents are the source of all magnetic fields. Two geometries are especially important: the long straight wire and the solenoid.

B = μ₀I / 2πrLong straight wire

B = μ₀nISolenoid (n = N/L)

μ₀ = 4π×10⁻⁷H/m (permeability)

Long Straight Wire

Current I (A)

10 A

Distance r (cm)

5 cm

40.0
B field (µT)

4.00×10⁻⁵
B field (T)

Solenoid

Turns N

500 turns

Length L (cm)

20 cm

Current I (A)

2 A

2500
n (turns/m)

6.28
B field (mT)

Earth's field~50 µT — typical long-wire result at a few cm

MRI scanner1.5–3 T — thousands of turns with superconducting wire

Right-hand ruleThumb along I → fingers curl in the direction of B rings

Lesson 15 — Electromagnetic Induction

Faraday's Law & Lenz's Law

A changing magnetic flux through a loop induces an EMF. This is the principle behind every generator and transformer — changing magnetism creates electricity.

Φ = BAcosθMagnetic flux (Wb)

EMF = −NΔΦ/ΔtFaraday's Law

Lenz's LawInduced I opposes ΔΦ

Magnetic Flux

Field B (mT)

100 mT

Loop area A (cm²)

50 cm²

Angle θ (°)

0°

500
Flux Φ (µWb)

5.00×10⁻⁴
Flux Φ (Wb)

Induced EMF (Faraday's Law)

Number of turns N

100 turns

ΔΦ change (µWb)

200 µWb

Time Δt (ms)

10 ms

2.00
Induced EMF (V)

20.0
ΔΦ/Δt (Wb/s)

🔄Lenz's LawThe induced current always creates a field that opposes the change in flux — nature's resistance to change

⚡GeneratorRotating coil in B field → continuously changing Φ → AC voltage

🔁TransformerAC in primary coil → changing B → EMF in secondary (V ratio = N ratio)

🚂Eddy currentsInduction in solid conductors — used in braking, induction cooktops, metal detectors

θ = 0°B perpendicular to loop plane → maximum flux Φ = BA

θ = 90°B parallel to loop plane → zero flux (field lines don't pass through)

Faster changeSmaller Δt → larger EMF — why generators spin fast

What is a Circuit?

Ohm's Law

Series Circuits

Parallel Circuits

Electrical Power & Energy

Voltage Divider

Electric Potential

Equivalent Resistance

( R1 ∥ R2 ) + R3

R1 + ( R2 ∥ (R3 + R4) )

(R1 + R2) ∥ (R3 + R4)

Coulomb's Law

Electric Field

Potential from a Point Charge

Capacitance & Capacitors

Lorentz Force

Creating Magnetic Fields

Faraday's Law & Lenz's Law

Simulation Lab

Series + Parallel

Build Your Circuit