Electromagnetism: The Principle Map

Use this reference map to compare field-force, potential-flux-energy, device-network, and induction-wave relations across electric systems and magnetic or coupled systems.

Practice recalling the principles

Layer 1Layer 2Layer 3

Expand

Layer 1Layer 2Layer 3

− 100% + Fit

Close

On this page

• Why Learn Electromagnetism?
• Prerequisites
• The Principle Map
• Course-Coverage Boundary
• Retrieval Practice
• Core Principles
• What’s Next?
• How This Fits in Unisium

Why Learn Electromagnetism?

Electromagnetism is where physics stops looking like only contact forces and trajectories and starts looking like fields, potentials, flux, and circuit relations. Charges act at a distance through fields. Energy can be tracked through potential difference. Current, resistance, and power become a network story. Magnetic and induction laws add direction, orientation, and field coupling on top of that.

That makes EM structurally different from a formula sheet. The hard part is not just memorizing more equations. It is seeing what kind of physical regime and relation you are dealing with: electric versus magnetic or coupled systems, and within those, whether the step is about field and force, potential and flux, devices and networks, or induction and wave structure. A useful subdomain guide therefore has to do more than list formulas. It has to show how those relations group naturally and form a coherent learning path.

This map covers the core relations of a calculus-based introductory university course, including continuous sources, magnetic dipoles, inductors, transients, AC impedance, field energy, and electromagnetic-wave energy transport. It is a guide to the recurring relations and their conditions, not a claim that the equations replace the geometry, orientation, or physical reasoning needed to solve full problems.

Prerequisites

Mathematics:

• Algebra fluency: rearranging equations, handling proportional relations, and reading reciprocal structure
• Basic vector fluency: magnitude versus vector form, components, dot product, and cross-product direction language
• Comfort with functions and rates, especially for reading field and potential relations

Prior Subdomains:

• Algebra
• Functions
• Classical Mechanics

Calculus note: the map includes field integrals, line and surface orientation, continuous-source geometry, time derivatives, and local material relations. Learners should be comfortable with single-variable calculus and should expect multivariable notation to appear where the physical geometry requires it.

The Principle Map

The map organizes Electromagnetism along two axes:

X-axis (Relation family):

• Field and force - relations that connect charges, currents, fields, and forces
• Potential, energy, and flux - scalar potential structure, potential-energy structure, and field-through-surface relations
• Devices and networks - constitutive and bookkeeping relations for capacitors and lumped circuits
• Induction and waves - later coupled-field relations such as motional EMF and electromagnetic wave speed

Y-axis (Physical regime):

• Electric systems - electrostatics, electric potential and energy, electric flux, capacitors, and circuit-network relations
• Magnetic and coupled systems - magnetic force and field relations, magnetic flux, motional induction, and later wave structure

Layers:

• Core / algebraic - direct algebraic and geometry-constrained EM relations
• Integral and source-geometry field laws - flux integrals, closed-loop and closed-surface laws, field-potential calculus links, density-to-source-element setup, and source-distribution integrals
• Dynamic, local, and material models - time-varying circuit relations, phasor impedance relations, local current-field relations, material-response relations, field-energy relations, and EM-wave energy transport

Progression numbers provide one recommended route through the 63 principles. They do not claim that every course teaches electromagnetism in exactly this order.

The progression is a navigation aid. Use the relation family and physical regime to choose a nearby principle when your course follows a different order.

Why the map uses these axes

The horizontal axis follows the conceptual relation families that recur in electromagnetism. Scan left to right through field and force, potential and flux, devices and networks, and induction and waves. The vertical split then shows whether each relation belongs primarily to electric systems or to magnetic and coupled systems.

Some cells are naturally sparse. Electric systems contribute fewer standalone induction-and-wave relations, while device-and-network relations are more numerous in the electric lane than in the magnetic lane. That imbalance reflects the subject rather than a missing formula list.

The grid is most useful when it helps you distinguish equations that look similar but answer different physical questions. Conditions in the tables make surface choice, loop orientation, source geometry, sign conventions, and circuit regime explicit.

Course-Coverage Boundary

The principle map is broad enough to represent the core relations of a serious calculus-based introductory university electromagnetism course, often called Physics II or introductory E&M.

It includes normal late-intro material: continuous charge sources, Ampere-law magnetostatics, inductors, RC/RL transients, AC impedance, magnetic dipoles and current-loop torque, field energy, EM waves, and the Poynting vector.

It intentionally does not include the upper-division or honors field-theory sequence: differential Maxwell forms, Poisson and Laplace electrostatics, vector potential and gauge material, richer polarization and magnetization theory, transmission lines, waveguides, radiation theory, or full engineering electromagnetics. More advanced courses cover those topics, but they are not required for a strong introductory university E&M foundation.

Practice recalling the principles

This flashcard tool helps make the 63 Electromagnetism principles stronger and easier to access from memory. You practice recalling each principle’s equation or relation and its conditions before revealing the answer.

Use it if you often recognize an electromagnetism equation after seeing it, but struggle to choose between similar field, potential, flux, circuit, or induction relations. Start with core electric-system principles, then add magnetic and coupled systems or later layers when you want a broader review.

Why this works

This tool is based on Vegard Gjerde’s research on structured retrieval practice of physics principle structures. Across this line of work, students practiced retrieving named principles, equations, and conditions instead of only reviewing the completed principle structure. The broader strategy is explained in the Retrieval Practice guide.

Key papers:

• Gjerde, Holst, & Kolstø (2020). Retrieval practice of a hierarchical principle structure in university introductory physics: Making stronger students. Physical Review Physics Education Research, 16(1), 013103.
• Gjerde, Havre Paulsen, Holst, & Kolstø (2022). Problem solving in basic physics: Effective self-explanations based on four elements with support from retrieval practice. Physical Review Physics Education Research, 18(1), 010136.
• Gjerde, Marisaldi, Oksavik, Olafsson, Spångberg, & Holst (2025). Mandatory retrieval test to incentivize retrieval practice of physics principles. Physical Review Physics Education Research, 21(1), 010119.

Relation family

Field and force Potential, energy, and flux Devices and networks Induction and waves

Physical regime

Electric systems Magnetic and coupled systems

Order

Progression Random

Layers

Core / algebraic Integral field laws Dynamic, local, and material models

Start practice

Core Principles

The tables contain all 63 principles in the map.

Conditions tell you when a principle applies. They are intentionally concise here. Think of them as the discriminating assumptions that separate a valid EM relation from a tempting misuse.

Layer 2 or layer 3 does not mean less important. The layers distinguish the mathematical and physical demands around a relation, not a simple difficulty ranking.

Electric Systems: Field and Force

Principle	Equation	Condition
Coulomb Force	$F = k \frac{\lvert q_1 q_2 \rvert}{r^2}$	point charges; electrostatic; single medium; $k=\mathrm{const}$
Electric Field-Force Relation	$\vec{F} = q\vec{E}$	nonzero charge when solving for field; field evaluated at a point
Electric Field From Point Charge	$E = k \frac{\lvert q \rvert}{r^2}$	point charge; electrostatic; single medium; $k=\mathrm{const}$
Electric Field Superposition	$\vec{E}_{net} = \sum_i \vec{E}_i$	multiple sources; linear superposition regime
Electric Field From Continuous Charge Distribution	$\vec{E} = k\int \frac{dq}{r^2}\hat{r}$	continuous charge distribution; field point and source geometry defined

This is the natural starting point because the relations are direct even when geometry still creates problem difficulty. Coulomb Force and Electric Field From Point Charge use one magnitude equation in the table while their guides also explain the corresponding vector representation.

Electric Systems: Potential, Energy, and Flux

Principle	Equation	Condition
Electric Potential Of A Point Charge	$V = k \frac{q}{r}$	point charge; electrostatic; $k=\mathrm{const}$; reference fixed
Electric Potential Energy Of Two Point Charges	$U = k \frac{q_1 q_2}{r}$	point charges; electrostatic; $k=\mathrm{const}$; reference fixed
Electric Potential Energy From Potential	$\Delta U = q\Delta V$	charge in a region with defined potential difference
Uniform-Field Potential Difference	$\Delta V = -E\Delta x$	uniform field; signed displacement along field axis
Electric Flux In A Uniform Field	$\Phi_E = \vec{E} \cdot \vec{A}$	uniform field over surface; area vector defined
Electric Flux Integral	$\Phi_E = \int \vec{E} \cdot d\vec{A}$	surface and area orientation defined
Gauss Law	$\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$	closed surface; enclosed charge defined; outward orientation
Electric Potential Line Integral	$\Delta V = -\int_A^B \vec{E} \cdot d\vec{\ell}$	path endpoints and direction fixed; electrostatic field
Electric Field From Potential Gradient	$\vec{E} = -\nabla V$	differentiable potential field; coordinates fixed
Charge Density Differential Relation	$dq = \rho dV$	continuous charge model; density type and source element chosen
Electric Potential From Continuous Charge Distribution	$V = k\int \frac{dq}{r}$	continuous charge distribution; field point and source geometry fixed
Electric Field Energy Density	$u_E = \frac{1}{2}\epsilon E^2$	electric field magnitude defined; linear medium or vacuum permittivity chosen

This row is broader than scalar potential alone on purpose. It groups the relations that connect field description to scalar energy and surface-through-field structure. For the integral relations, surface choice and path or area orientation are part of applying the equation correctly.

Electric Systems: Devices and Networks

Principle	Equation	Condition
Capacitance Definition	$C = \frac{Q}{\Delta V}$	lumped-capacitance model
Parallel-Plate Capacitance	$C = \epsilon \frac{A}{d}$	parallel plates; negligible fringing; uniform medium
Capacitor Energy	$U = \frac{1}{2} C (\Delta V)^2$	capacitor with defined $C$ and $\Delta V$
Equivalent Capacitance In Series	$\frac{1}{C_{eq}} = \sum_i \frac{1}{C_i}$	series topology already identified
Equivalent Capacitance In Parallel	$C_{eq} = \sum_i C_i$	parallel topology already identified
Electric Current Definition	$I = \frac{\Delta Q}{\Delta t}$	charge flow through a surface; time interval defined
Resistance From Geometry	$R = \rho \frac{L}{A}$	uniform conductor; length and cross-section defined
Ohm’s Law	$\Delta V = IR$	ohmic element; lumped-circuit model; transient operation allowed
Electric Power	$P = I\Delta V$	lumped element; current and potential difference defined
Equivalent Resistance In Series	$R_{eq} = \sum_i R_i$	series topology already identified
Equivalent Resistance In Parallel	$\frac{1}{R_{eq}} = \sum_i \frac{1}{R_i}$	parallel topology already identified
Kirchhoff Junction Rule	$\sum I_{in} = \sum I_{out}$	lumped-circuit model; steady current bookkeeping
Kirchhoff Loop Rule	$\sum \Delta V = 0$	closed loop chosen; sign convention fixed
Current Density Definition	$I = \int \vec{J}\cdot d\vec{A}$	current distribution and oriented surface defined
Microscopic Ohm’s Law	$\vec{J} = \sigma\vec{E}$	ohmic material; local field and conductivity defined
Capacitor Time Constant	$\tau = RC$	first-order RC circuit; effective resistance and capacitance identified
RC Charging Voltage	$V_{C}(t)=V_s(1-e^{-t/RC})$	series RC step charging; $V_C(0)=0$
RC Discharging Voltage	$V_{C}(t)=V_{0}e^{-t/RC}$	series RC discharge path; initial capacitor voltage specified
Resistor Impedance	$Z_R = R$	sinusoidal steady-state; phasor convention and $R$ defined
Capacitor Impedance	$Z_C = \frac{1}{i\omega C}$	sinusoidal steady-state; phasor convention/$C$/$\omega$ defined

This combined cell is much more honest than separate capacitor and circuit columns. It gathers the device and network relations that students use once EM stops being only field-at-a-point reasoning and starts becoming component and loop bookkeeping. It also makes the real tension visible: the relations are honest, but topology recognition and sign bookkeeping still live upstream of several of them.

Magnetic and Coupled Systems: Field and Force

Principle	Equation	Condition
Magnetic Force On A Moving Charge	$\vec{F} = q\vec{v} \times \vec{B}$	moving charge in a magnetic field
Lorentz Force	$\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$	charge in defined electric and magnetic fields
Magnetic Force On A Wire	$\vec{F} = I\vec{L} \times \vec{B}$	straight current-carrying segment in a magnetic field
Magnetic Field Near A Long Straight Wire	$B = \frac{\mu_0 I}{2\pi r}$	long straight wire; steady current; point outside wire
Magnetic Field In A Long Solenoid	$B = \mu_0 n I$	long solenoid; interior field approximation
Biot-Savart Law	$\vec{B} = \frac{\mu_0}{4\pi}\int \frac{I d\vec{\ell} \times \hat{r}}{r^2}$	steady current distribution; source geometry defined
Ampere Law	$\oint \vec{B}\cdot d\vec{\ell} = \mu_0 I_{enc}$	magnetostatic regime; closed loop; enclosed current defined
Torque On A Magnetic Dipole	$\vec{\tau}=\vec{\mu}\times\vec{B}$	magnetic dipole in a magnetic field; orientation defined

This cell is the magnetic analog of the electric field-and-force group. It connects force laws for moving charges and current-carrying wires with the magnetic-field relations that commonly feed them. The vector equations keep direction visible, while the familiar magnitude forms remain useful when the geometry is already understood. Lorentz Force combines electric and magnetic effects, while Biot-Savart Law connects a current distribution to its magnetic field and therefore depends strongly on geometry.

Magnetic and Coupled Systems: Potential, Energy, and Flux

Principle	Equation	Condition
Magnetic Flux In A Uniform Field	$\Phi_B = \vec{B} \cdot \vec{A}$	uniform field over surface; area vector defined
Magnetic Flux Integral	$\Phi_B = \int \vec{B} \cdot d\vec{A}$	surface and area orientation defined
Gauss Law For Magnetism	$\oint \vec{B} \cdot d\vec{A} = 0$	closed surface; outward area orientation
Inductor Energy	$U = \frac{1}{2}LI^2$	inductor model with defined inductance and current
Magnetic Field Energy Density	$u_B = \frac{B^2}{2\mu}$	magnetic field magnitude defined; linear medium or vacuum permeability chosen
Magnetic Dipole Energy	$U=-\vec{\mu}\cdot\vec{B}$	magnetic dipole in uniform external field; reference fixed

This cell is smaller in the core layer, but the field-calculus layer names the natural magnetic-flux and no-monopole relations explicitly. Surface orientation remains essential when applying them.

Magnetic and Coupled Systems: Devices and Networks

Principle	Equation	Condition
Inductance-Flux Relation	$N\Phi_B = LI$	inductor or coil model; flux linkage and current defined
Inductor Voltage Relation	$\Delta V_L = L\frac{dI}{dt}$	inductor model; passive sign convention fixed
RL Time Constant	$\tau = \frac{L}{R}$	first-order RL circuit; effective resistance and inductance identified
Inductor Impedance	$Z_L = i\omega L$	sinusoidal steady-state; phasor convention/$L$/$\omega$ defined
Magnetic Dipole Moment Of A Current Loop	$\vec{\mu}=NI\vec{A}$	planar current loop; turns/current/area vector defined

This magnetic device lane includes inductors, RL time scales, AC inductor impedance, and current-loop dipole moment. Sign convention, area-vector orientation, and transient-circuit framing determine which form applies.

Magnetic and Coupled Systems: Induction and Waves

Principle	Equation	Condition
Motional EMF	$\mathcal{E} = BLv$	standard motional-EMF geometry; perpendicular motion resolved
Faraday Law Finite Change	$\mathcal{E}_{avg} = -\frac{\Delta \Phi_B}{\Delta t}$	loop orientation fixed; average over time interval
Electromagnetic Wave Speed	$c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$	vacuum-wave context
Faraday Law Integral	$\oint \vec{E} \cdot d\vec{\ell} = -\frac{d\Phi_B}{dt}$	closed loop; loop orientation fixed
Ampere-Maxwell Law	$\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{enc} + \mu_0\epsilon_0\frac{d\Phi_E}{dt}$	closed loop; enclosed current and electric-flux change defined
Electromagnetic Wave Field Relation	$E = cB$	plane electromagnetic wave; same point and time
Poynting Vector Definition	$\vec{S} = \frac{1}{\mu}\vec{E}\times\vec{B}$	co-located fields; linear medium convention fixed

This group makes the induction and Maxwell bridge explicit. It also highlights decisions that the equation alone cannot make: loop orientation, sign conventions, enclosed-current choices, and the interpretation of changing flux.

Where to Go Next

Use the linked principle guides to move from the map to worked examples and retrieval practice. A practical sequence is:

1. Start with Electric Systems: Field and Force to establish the charge-force-field lane.
2. Move to Electric Systems: Potential, Energy, and Flux so scalar and surface relations sit next to the earlier field picture.
3. Treat Electric Systems: Devices and Networks as the first component and circuit lane rather than as isolated textbook fragments.
4. Use the magnetic and coupled-system rows once cross products, magnetic flux, and induction enter the picture.

Later expansion lanes outside this introductory map:

• differential Maxwell forms, Poisson/Laplace electrostatics, vector potential, advanced media, transmission lines, waveguides, radiation theory, and other field-theory or engineering-electromagnetics relations

How This Fits in Unisium

This map connects each principle to the broader electromagnetism structure. It keeps direction rules, symmetry choices, topology recognition, and multi-step problem strategies visible alongside the equations. Use the map to locate the relation you need, then use its guide to learn the conditions, representations, examples, and common failure modes.

Masterful Learning

The book behind these guides: a study system for physics, math, & programming built on retrieval, connection, explanation, and problem solving.

About the book →