Courses › PHYS 210 + PHYS 210L

About this course

This is a free self-study course outline written by Saving Planets. It is not a university course, it carries no academic credit, and no school is involved.

Its scope follows the second semester of calculus-based college physics (electricity and magnetism, with a first look at optics). It is modeled on the topics and depth of MIT OpenCourseWare 8.02, Caltech’s Ph1b and Ph1c, Georgia Tech PHYS 2212 and the free OpenStax University Physics Volume 2, which we link throughout. None of those authors or schools endorses this page.

What we add: every unit ties to real clean-energy or Earth-observation work with a cited public source (generators, solar panels, electric cars, the power grid, Earth’s magnetic field, space weather, satellite sensors), every number on the page is re-computed by a checking script before it goes live, and every lab has a free PhET version plus a safe low-voltage home version.

Work through it in order. Each unit has goals, key ideas, equations, step-by-step examples, practice with full answers, common mistakes, flashcards and a quiz. Your progress is saved only in this browser.

How this course helps save the planet

Almost all clean electricity is electromagnetism at work. Wind and hydro turbines spin generators that run on Faraday’s law, solar panels push charge with built-in electric fields, electric cars turn current into motion with magnetic forces, and the grid uses transformers and high voltage to move power with small losses.

Electromagnetic waves are also how we watch the planet: satellites such as Landsat 9 and GOES-19 read visible, infrared and microwave light to map crops, forests, water and storms. And Earth’s own magnetic field shields us, while space weather from the Sun can push currents into power lines. Each unit ends with a short, sourced section. Values that are not from a source are labeled as sample values.

Before you start

Prerequisites

Physics I: forces, energy, momentum, circular motion and oscillations.
Calculus I and II: derivatives, integrals, and exponential growth and decay.
Vectors: components, dot and cross products (taught briefly where needed).
Calculus III helps with Gauss’s and Ampère’s laws in integral form but is not required.

How to study

Draw the field or the circuit first. Most E&M mistakes are sign or direction mistakes.
Solve with symbols first, then put in numbers in SI units, and check the units of the answer.
Use symmetry: before applying Gauss’s or Ampère’s law, ask which way the field must point and on what it can depend.
Do every practice problem on paper before opening the answer.
Predict, then test it in the PhET simulation. A wrong prediction you fix sticks better than a right guess.
Retake the quizzes a few days later. Spaced practice beats one long session.

Lab equipment (for the wet labs)

A computer or tablet for the free PhET simulations (they run in a browser).
A digital multimeter and a few AA batteries with a battery holder (low voltage only).
A small set of resistors (100 Ω to 10 kΩ), a 100 µF or larger electrolytic capacitor rated above the battery voltage, jumper wires or a breadboard.
A smartphone with a magnetometer app (most phones have the sensor), a small compass, a few fridge magnets.
A clear glass or plastic box of water, a laser pointer of class 2 or lower or a narrow flashlight beam, a protractor.
Spreadsheet software for graphs and fits.

Suggested 15-week schedule

About 6 to 9 hours a week: reading, practice problems and, in lab weeks, the lab.

Week	Unit	Topics	Reading
1	Charge, Coulomb’s Law and Electric Fields	Charge, Coulomb’s law, superposition	OpenStax UP2 5.1–5.3
2	Charge, Coulomb’s Law and Electric Fields	Electric fields, continuous charge, field lines; Lab A	UP2 5.4–5.6
3	Gauss’s Law	Flux and Gauss’s law; conductors	UP2 6.1–6.4
4	Electric Potential and Capacitance	Potential energy, potential, equipotentials	UP2 7.1–7.5
5	Electric Potential and Capacitance	Capacitance, energy, dielectrics; Lab B	UP2 8.1–8.4
6	Current, Resistance and DC Circuits	Current, resistance, power	UP2 9.1–9.5
7	Current, Resistance and DC Circuits	DC circuits, Kirchhoff’s rules, RC circuits; Lab C	UP2 10.1–10.5
8	Magnetic Fields and Forces	Magnetic force, particle orbits, motors; midterm practice	UP2 11.1–11.5
9	Sources of Magnetic Fields: Biot–Savart and Ampère	Biot–Savart, Ampère’s law, solenoids; Lab D	UP2 12.1–12.7
10	Electromagnetic Induction: Faraday and Lenz	Faraday’s and Lenz’s laws, generators; Lab E	UP2 13.1–13.7
11	Inductance, RL and LC Circuits	Inductance, RL and LC circuits	UP2 14.1–14.6
12	AC Circuits, Transformers and the Power Grid	AC circuits, resonance, power, transformers	UP2 15.1–15.6
13	Maxwell’s Equations, Electromagnetic Waves and Optics	Maxwell’s equations, EM waves, energy and the spectrum	UP2 16.1–16.5
14	Maxwell’s Equations, Electromagnetic Waves and Optics	Optics: reflection, refraction, polarization, lenses; Lab F	UP3 1.1–1.7, 2.4
15	Review	Review and final practice set	All units

Your progress

Tick units and labs as you finish them, and take the quizzes. Progress and quiz scores are saved only in this browser, never sent to us.

Turn on JavaScript to track progress.

The lab (PHYS 210L)

The lab is the hands-on part of the course, a 1-credit component in a typical college schedule. Here it is free and for practice only.

Every lab can be done fully in a free PhET simulation, and most also have a home version that uses only batteries of 9 V or less. Never use wall power, never open a device that plugs into the wall, and never short a battery.

Safety reference: OSHA: Electrical safety

Lab A: Mapping electric fields and equipotentials

Week 2

Goal: Measure how the field of a point charge falls with distance, confirm the inverse-square law with a log-log fit, and map equipotentials around a dipole.

E=\frac{kq}{r^2}\ \Rightarrow\ \log E=\log(kq)-2\log r

A straight line of slope −2 on a log-log plot

Procedure

Open PhET Charges and Fields. Place a single +1 nC charge and turn on the values and grid.
Use the E-field sensor at 0.5, 1.0, 1.5 and 2.0 m and record E.
Plot log E against log r and fit a line; compare the slope with −2 and the intercept with log(kq).
Add a −1 nC charge 1 m away, use the voltmeter tool to trace the 0 V and ±10 V equipotentials, and sketch field lines perpendicular to them.

Data table

r (m)	E (V/m)
0.5	35.95
1.0	8.99
1.5	3.99
2.0	2.25

Values computed from E = kq/r² for q = 1 nC; your simulation readings should match to within rounding.

Sample calculation

From 0.5 m to 2.0 m (×4 in r), E falls from 35.95 to 2.25 V/m (×1/16), so the log-log slope is −2.
The 0 V equipotential of the dipole is the perpendicular bisector of the two charges, yet the field there is not zero.

Safety

None beyond normal computer use.

Home or virtual alternative

Rub a balloon on hair and watch it lift paper bits at different distances: a qualitative look at a field that falls off quickly with distance.

PhET: Charges and Fields · PhET: Coulomb’s Law

Planet link: Field maps like these are how engineers design the electric fields inside solar cells and high-voltage equipment. U.S. DOE: How does solar work?

I completed this lab (wet or virtual)

Lab B: Capacitance and plate separation

Week 5

Goal: Measure how a parallel-plate capacitor’s capacitance depends on plate separation, and extract ε₀ from the data.

C=\frac{\varepsilon_0A}{d}

Plot C against 1/d; the slope is ε₀A

Procedure

Open PhET Capacitor Lab: Basics. Set the plate area to 400 mm² and connect the battery at 1.5 V.
Read C at plate separations of 2, 4, 6, 8 and 10 mm.
Plot C against 1/d and fit a line through the origin; divide the slope by A to get ε₀.
At d = 5 mm, record the stored energy and compare with ½CV².

Data table

d (mm)	C (pF)
2	1.77
4	0.885
6	0.590
8	0.443
10	0.354

Values computed from C = ε₀A/d; simulation readouts round them.

Sample calculation

C × d is constant: 1.77 pF × 2 mm = 0.354 pF × 10 mm, so C ∝ 1/d.
Slope ε₀A = 3.54 × 10⁻¹⁵ F·m, so ε₀ = 3.54 × 10⁻¹⁵/(4 × 10⁻⁴) = 8.85 × 10⁻¹² F/m.
At 5 mm, C = 0.708 pF and U = ½CV² = 0.797 pJ at 1.5 V.

Safety

None beyond normal computer use.

Home or virtual alternative

Optional: with a multimeter that reads capacitance, measure two pieces of aluminum foil separated by one, two and three sheets of paper. C should fall as the stack grows.

PhET: Capacitor Lab: Basics

Planet link: Capacitors smooth power in electric cars, solar inverters and the grid; the same C = κε₀A/d sets their size. FuelEconomy.gov (DOE/EPA): All-electric vehicles

I completed this lab (wet or virtual)

Lab C: Ohm’s law, Kirchhoff’s rules and an RC time constant

Week 7

Goal: Confirm Ohm’s law, check Kirchhoff’s rules in a two-loop circuit, and measure the time constant of a charging capacitor.

V_C=V_0\left(1-e^{-t/RC}\right)

Charging curve; ln(1 − V_C/V₀) = −t/RC is a straight line

Procedure

Ohm’s law: put a 1 kΩ resistor across 1.5, 3.0, 4.5, 6.0 and 9.0 V (AA cells in series, or the PhET battery) and record the current; plot I against V.
Kirchhoff: in PhET Circuit Construction Kit: DC, build the Unit 4 circuit (12 V, 6 V, 2 Ω, 3 Ω, 6 Ω) and measure the three currents; check the junction and loop sums.
RC: charge a 100 µF capacitor from 9 V through 10 kΩ, reading V_C every second; plot ln(1 − V_C/9) against t and find RC from the slope.

Data table

t (s)	V_C (V), 10 kΩ and 100 µF from 9 V
0	0.00
1	5.69
2	7.78
3	8.55
5	8.94

Values computed for ideal parts; real capacitors are often ±20% from their label, so your τ may differ.

Sample calculation

Ohm’s law: the slope of I against V is 1 mA/V, so R = 1.00 kΩ; at 9 V the resistor dissipates only 81 mW.
Kirchhoff: 2 A − 0.667 A = 1.33 A at the junction, and 12 − 2(2) − 6(1.33) = 0 around loop 1.
RC: the line through ln(1 − V_C/9) has slope −1.0 s⁻¹, so τ = 1.0 s = (10 kΩ)(100 µF).

Safety

Use only batteries of 9 V or less; never wall power.
Never connect a wire straight across a battery: it heats fast and can burn.
Electrolytic capacitors have a + lead; connect it toward the battery’s + side, and use one rated above 9 V.

Home or virtual alternative

Everything here works with a multimeter, AA cells, a 9 V battery, resistors and a capacitor on a breadboard; time the RC readings with a phone video of the meter.

PhET: Circuit Construction Kit: DC · PhET: Ohm’s Law · PhET: Resistance in a Wire

Planet link: Solar arrays are wired by the same rules: series strings raise voltage and parallel strings raise current. U.S. DOE: How does solar work?

I completed this lab (wet or virtual)

Lab D: Measuring magnetic fields, including Earth’s

Week 9

Goal: Measure Earth’s magnetic field with a phone magnetometer, map the field of a wire in simulation, and compare both with theory.

B=\frac{\mu_0I}{2\pi r}

Field of a long wire

Procedure

Open a magnetometer app away from metal and electronics. Record the total field magnitude at five spots and average them.
Compare with NOAA’s stated range of about 25,000–65,000 nT at Earth’s surface.
In PhET Magnets and Electromagnets, see how the field changes with current and number of loops; then compute B for a long wire with I = 2 A at r = 1, 2 and 4 cm and plot B against 1/r.
Bring a fridge magnet toward the phone and note the distance at which its field equals Earth’s.

Data table

r (cm)	B for a 2 A wire (µT)
1	40
2	20
4	10

Wire values computed from μ₀I/(2πr). Earth-field readings depend on your location; NOAA’s calculator gives the expected total for your spot.

Sample calculation

B ∝ 1/r: doubling the distance halves the field, so the plot of B against 1/r is a straight line through the origin with slope μ₀I/(2π) = 0.4 µT·m.
A 2 A wire matches Earth’s 50 µT field at r = μ₀I/(2π × 50 µT) = 0.8 cm.

Safety

Keep strong magnets away from phones’ card slots, pacemakers and credit cards.
Do not run high current through bare wire at home; the wire part of this lab is done in simulation.

Home or virtual alternative

The magnetometer and fridge-magnet steps are the home version.

PhET: Magnets and Electromagnets · PhET: Magnet and Compass · NOAA magnetic field calculators

Planet link: Knowing the normal field is the baseline for spotting geomagnetic storms, which NOAA warns can induce harmful currents in power grids. NOAA NCEI: Geomagnetism frequently asked questions · NOAA Space Weather Prediction Center: Geomagnetic storms

I completed this lab (wet or virtual)

Lab E: Induction and a model generator

Week 10

Goal: Show that induced EMF depends on the rate of change of flux, and that a generator’s peak EMF grows in proportion to its spin rate.

\varepsilon_{\max}=NBA\omega

Peak EMF of a spinning coil

Procedure

In PhET Faraday’s Law, push the magnet through the coil slowly and then quickly; record the peak voltage reading each time and note the sign change when it leaves.
In PhET Faraday’s Electromagnetic Lab (Generator tab), change the turbine speed and record the peak bulb or voltmeter response.
Compute ε_max for a model coil with N = 200, A = 0.01 m² and B = 0.05 T at 1, 2 and 3 revolutions per second.
Home: wind 100 turns of thin insulated wire around a toilet-paper tube, connect a multimeter on mV, and drop a strong magnet through it.

Data table

Spin rate (rev/s)	Peak EMF (V)
1	0.628
2	1.26
3	1.88

Values computed for the model coil (sample values).

Sample calculation

ε_max = NBA(2πf) = 200 × 0.05 × 0.01 × 2π f = 0.628 V per rev/s, so doubling the speed doubles the EMF.
A faster magnet gives a taller, shorter pulse; the area under the EMF-time curve (total flux change × N) stays the same.

Safety

Strong neodymium magnets snap together hard and can pinch skin or shatter; handle one at a time.
Keep magnets away from pacemakers and electronics.

Home or virtual alternative

The coil-and-magnet step is the home version; any multimeter on its millivolt range works.

PhET: Faraday’s Law · PhET: Faraday’s Electromagnetic Lab · PhET: Generator

Planet link: Every wind turbine and hydro plant runs this lab at megawatt scale: the turbine spins a generator to make electricity. U.S. DOE: How do wind turbines work? · U.S. DOE: How hydropower works

I completed this lab (wet or virtual)

Lab F: Refraction and Snell’s law

Week 14

Goal: Measure refraction angles from air into water, find the index of refraction, and observe total internal reflection.

n=\frac{\sin\theta_1}{\sin\theta_2}

Slope of sin θ₁ against sin θ₂

Procedure

In PhET Bending Light, set the lower medium to water and shine the laser at 10° to 60° from the normal in 10° steps; read the refraction angle with the protractor tool.
Plot sin θ₁ against sin θ₂ and fit a line through the origin; the slope is n.
Put the laser in the water and increase the angle until the refracted beam vanishes; record the critical angle.
Home: shine a low-power laser or narrow flashlight beam into a clear box of water with a drop of milk and measure the angles on a printed protractor.

Data table

θ₁ in air (°)	θ₂ in water (°)
10	7.5
20	14.9
30	22.1
40	28.9
50	35.2
60	40.6

Values computed from Snell’s law with n = 1.33.

Sample calculation

sin 30°/sin 22.1° = 0.500/0.376 = 1.33, and every row gives the same ratio.
Critical angle from water to air: sin⁻¹(1/1.33) = 48.8°.

Safety

Use only a class 2 (under 1 mW) laser pointer, and never look into the beam or aim it at anyone’s eyes; watch for reflections off the water and glass.

Home or virtual alternative

The water-box step is the home version.

PhET: Bending Light

Planet link: Satellite sensors are built from lenses, mirrors and filters designed with these same laws of refraction and reflection. NASA Landsat Science: Landsat 9 instruments and spectral bands

I completed this lab (wet or virtual)

Exam prep practice sets

Time yourself: about 10 minutes per problem. Then compare with the full solutions.

Midterm practice set (Units 1–5)

Charges of +2.0 µC and +8.0 µC are 0.30 m apart. Find the force between them and the point between them where E = 0.
Show full solution
1. F = k(2 × 10⁻⁶)(8 × 10⁻⁶)/0.09.
2. k(2)/x² = k(8)/(0.3 − x)² gives x = 0.10 m.
Answer: 1.60 N, repulsive; 0.10 m from the 2.0 µC charge.
A closed surface surrounds a 3.0 nC charge. Find the net flux.
Show full solution
1. Φ = Q/ε₀.
Answer: ≈ 339 N·m²/C.
A 10 µF capacitor is charged to 12 V. Find Q and U.
Show full solution
1. Q = CV; U = ½CV².
Answer: 120 µC; 0.72 mJ.
A 9 V battery drives 3 Ω in series with a parallel pair of 6 Ω and 3 Ω. Find the current and total power.
Show full solution
1. Parallel pair = 2 Ω; total 5 Ω.
2. I = 9/5; P = IV.
Answer: 1.8 A; 16.2 W.
A proton moves at 1.0 × 10⁶ m/s across a 0.20 T field. Find the orbit radius.
Show full solution
1. r = mv/(eB).
Answer: ≈ 5.22 cm.

Final practice set (all units)

Find B 0.10 m from a long wire carrying 50 A.
Show full solution
1. B = μ₀I/(2πr).
Answer: 100 µT, about twice Earth’s field.
A 50-turn coil of area 0.05 m² spins at 50 rev/s in a 0.20 T field. Find the peak EMF.
Show full solution
1. ε = NBAω, ω = 2π × 50.
Answer: ≈ 157 V.
An RL circuit has L = 0.5 H and R = 100 Ω. Find τ and the fraction of the final current after 5 ms.
Show full solution
1. τ = L/R.
Answer: 5 ms; 63%.
Send 50 MW at 230 kV over a line with 5 Ω resistance. Find the current and loss.
Show full solution
1. I = P/V; loss = I²R.
Answer: 217 A; 0.236 MW (0.47%).
Find the frequency and photon energy of 500 nm light.
Show full solution
1. f = c/λ; E = hf.
Answer: 6.00 × 10¹⁴ Hz; 2.48 eV.
Light enters glass (n = 1.5) at 45°. Find the refraction angle.
Show full solution
1. sin θ₂ = sin 45°/1.5.
Answer: 28.1°.

Formula sheet

F=k|q_1q_2|/r^2

\vec E=\vec F/q

\oint\vec E\cdot d\vec A=Q_{\text{enc}}/\varepsilon_0

V=kq/r,\ E_x=-dV/dx

C=\kappa\varepsilon_0A/d,\ U=\tfrac12CV^2

V=IR,\ R=\rho L/A,\ P=IV

\tau=RC,\ \tau=L/R

\vec F=q\vec v\times\vec B,\ r=mv/(qB)

\vec F=I\vec L\times\vec B,\ \tau=NIAB\sin\theta

B=\mu_0I/(2\pi r),\ B=\mu_0nI

\varepsilon=-N\,d\Phi_B/dt,\ \varepsilon=BLv

L=\mu_0N^2A/l,\ U=\tfrac12LI^2

X_L=\omega L,\ X_C=1/(\omega C),\ Z=\sqrt{R^2+(X_L-X_C)^2}

V_2/V_1=N_2/N_1

c=1/\sqrt{\mu_0\varepsilon_0}=f\lambda,\ E=hf

n_1\sin\theta_1=n_2\sin\theta_2,\ 1/f=1/d_o+1/d_i

Constants

Constant	Value
Elementary charge, e	1.602 × 10⁻¹⁹ C
Coulomb constant, k = 1/(4πε₀)	8.988 × 10⁹ N·m²/C²
Vacuum permittivity, ε₀	8.854 × 10⁻¹² F/m
Vacuum permeability, μ₀	1.257 × 10⁻⁶ T·m/A (≈ 4π × 10⁻⁷)
Speed of light, c	2.998 × 10⁸ m/s
Electron mass, mₑ	9.109 × 10⁻³¹ kg
Proton mass, mₚ	1.673 × 10⁻²⁷ kg
Planck constant, h	6.626 × 10⁻³⁴ J·s

Source: NIST: Fundamental physical constants

Free texts and simulations

Use these free textbooks, lectures and simulations alongside the course.

OpenStax University Physics Volume 2: Free, complete textbook for this course.
MIT OpenCourseWare 8.02 Physics II: Electricity and Magnetism: Free MIT lectures, problem sets and exams.
OpenStax University Physics Volume 3 (optics): Free optics chapters for Unit 10.
PhET: Charges and Fields: Place charges and map fields and equipotentials.
PhET: Circuit Construction Kit: DC: Build and measure circuits.
PhET: Faraday’s Electromagnetic Lab: Magnets, coils, generators and transformers.

Glossary

Ampère’s law: ∮B·dl = μ₀I_enc.
Capacitance: Charge stored per volt, C = Q/V.
Coulomb’s law: Force between point charges, F = k|q₁q₂|/r².
Critical angle: Angle beyond which light is totally reflected inside a denser medium.
Displacement current: ε₀ dΦ_E/dt, Maxwell’s addition to Ampère’s law.
EMF: Energy per charge supplied by a source such as a battery or generator.
Electric field: Force per unit charge.
Electric potential: Potential energy per unit charge, in volts.
Faraday’s law: Changing magnetic flux induces EMF.
Gauss’s law: Net electric flux out of a closed surface equals Q_enc/ε₀.
Geomagnetically induced current: Current driven in grids and pipelines by geomagnetic storms.
Impedance: Total opposition to AC, Z.
Inductance: EMF per rate of change of current, in henries.
Lenz’s law: Induced current opposes the change in flux.
Magnetic flux: Φ_B = ∫B·dA, in webers.
Maxwell’s equations: Four laws that unify electricity, magnetism and light.
Power factor: cos φ, the fraction of apparent power that does work.
RMS value: Root-mean-square, V₀/√2 for a sine wave.
Resistivity: Material property ρ in R = ρL/A.
Snell’s law: n₁ sin θ₁ = n₂ sin θ₂.
Transformer: Two coupled coils that change AC voltage.

Self-assessment rubric

No grade or credit is given. Use this to judge honestly where you are in each unit.

Level	What it looks like
4 · Mastery	Sets up fields and circuits from a description, chooses the right law, handles signs and directions, and checks units and limits.
3 · Proficient	Solves the practice problems with occasional sign or unit slips.
2 · Developing	Follows worked examples but struggles to choose a Gaussian surface, loop or right-hand rule.
1 · Beginning	Plugs numbers into formulas without a picture; needs to revisit the unit.

Suggested weighting if you want a score

Part	Weight
Practice problems	25%
Labs (PHYS 210L)	20%
Midterm practice set	25%
Final practice set	30%

Aim for level 3 or better in every unit before moving to the next course.

Physics II + Lab

About this course

How this course helps save the planet

Before you start

Prerequisites

How to study

Lab equipment (for the wet labs)

Suggested 15-week schedule

Your progress

Charge, Coulomb’s Law and Electric Fields

Learning objectives

Key concepts

Key equations

Worked examples, step by step

Practice problems

Common misconceptions

Flashcards

Self-check quiz

How this helps save the planet

Free learning resources

Gauss’s Law

Learning objectives

Key concepts

Key equations

Worked examples, step by step

Practice problems

Common misconceptions

Flashcards

Self-check quiz

How this helps save the planet

Free learning resources

Electric Potential and Capacitance

Learning objectives

Key concepts

Key equations

Worked examples, step by step

Practice problems

Common misconceptions

Flashcards

Self-check quiz

How this helps save the planet

Free learning resources

Current, Resistance and DC Circuits

Learning objectives

Key concepts

Key equations

Worked examples, step by step

Practice problems

Common misconceptions

Flashcards

Self-check quiz

How this helps save the planet

Free learning resources

Magnetic Fields and Forces

Learning objectives

Key concepts

Key equations

Worked examples, step by step

Practice problems

Common misconceptions

Flashcards

Self-check quiz

How this helps save the planet

Free learning resources

Sources of Magnetic Fields: Biot–Savart and Ampère

Learning objectives

Key concepts

Key equations

Worked examples, step by step

Practice problems

Common misconceptions

Flashcards

Self-check quiz

How this helps save the planet

Free learning resources

Electromagnetic Induction: Faraday and Lenz

Learning objectives

Key concepts

Key equations

Worked examples, step by step