You're staring at a blank page. The exam starts in forty minutes. Somewhere in your bag is a crumpled formula sheet you printed three weeks ago and never actually looked at And it works..
Sound familiar?
Here's the thing about E&M — electricity and magnetism — that nobody tells you in lecture: the equations aren't the hard part. It's knowing which one to reach for when the problem stops looking like the textbook examples and starts looking like a mess of vectors, integrals, and "wait, is that r-hat or r-vector?"
I've been there. So has every physics major who survived junior year. This isn't a reference sheet you memorize the night before. It's a map. And like any map, it's useless if you don't know the territory That alone is useful..
What Is an E&M Equation Sheet
At its core, an E&M equation sheet is a curated collection of the fundamental relationships governing electric and magnetic fields, forces, potentials, and induction. But that's the textbook definition Most people skip this — try not to. Worth knowing..
In practice? It's the one piece of paper you're allowed to bring into the exam that separates "I recognize this problem" from "I have no idea where to start."
Most sheets cover the same ground: Coulomb's law, Gauss's law, Ampère's law, Faraday's law, the Lorentz force, capacitance and inductance formulas, maybe a few vector identities. The difference between a useful sheet and a useless one comes down to organization, annotation, and — this is key — whether you built it yourself or downloaded someone else's Worth keeping that in mind..
The Two Types of Sheets
Course-provided sheets are standardized. Your professor hands them out or posts them on Canvas. They're clean, official, and everyone has the exact same one. Pros: you can't get in trouble for using it. Cons: it's often sparse, missing the derived forms you actually need, and you didn't make it — so you don't know where anything lives.
Self-made sheets take time. You condense a semester's worth of lectures, homework, and textbook derivations onto one or two pages. Pros: the act of making it is studying. You know exactly where Gauss's law in differential form sits because you wrote it there at 2 AM. Cons: if you leave something out, it's on you.
Most upper-division courses let you bring your own. Intro courses often don't. Worth adding: check the syllabus. Then check it again.
Why It Matters / Why People Care
E&M is the first physics course where intuition actively fights you. Mechanics? You've been catching balls and sliding down slides your whole life. Consider this: your gut mostly works. But you've never felt a curl of E or watched a B-field do work (spoiler: it doesn't) Practical, not theoretical..
The equation sheet matters because it externalizes the cognitive load. When you're three steps into a boundary value problem and your working memory is full of Legendre polynomials, you don't want to be deriving the potential of a dipole from scratch.
It also matters because E&M is connected. The same physics shows up in four different guises:
- Integral form (flux through surfaces, circulation around loops)
- Differential form (divergence and curl at a point)
- Potential form (scalar and vector potentials)
- Energy form (field energy density, Poynting vector)
A good sheet makes those connections visible. A bad sheet hides them behind a wall of symbols.
And let's be honest — partial credit lives on that sheet. In real terms, writing down the right starting equation, even if you botch the integral, can be the difference between a C and a B. In practice, i've seen it happen. I've been it.
How It Works (or How to Build One That Works)
Don't just copy equations. Organize them by physical situation, not by chapter number. Your brain retrieves by context: "infinite line charge" → "Gauss's law, cylindrical symmetry" → "E = λ/2πε₀r". Not "Chapter 2, Equation 14 The details matter here..
Electrostatics: The Foundation
Start with Coulomb. Not because you'll use it directly — you won't, not for continuous distributions — but because everything else derives from it.
Coulomb's Law (point charges):
F = k q₁q₂/r² r̂
E = k q/r² r̂
k = 1/4πε₀. Write both forms. ε₀ shows up in capacitance; k shows up in quick estimates. You'll want both.
Superposition Principle — this isn't an equation, it's a rule. Write it anyway: "Fields add as vectors. Potentials add as scalars." That one line saves more sign errors than anything else.
Gauss's Law — integral and differential, side by side:
∮ E·dA = Q_enc/ε₀
∇·E = ρ/ε₀
Below each, list the three symmetries where the integral form actually works: spherical, cylindrical, planar. Add the results:
- Point charge / sphere: E = Q/4πε₀r²
- Infinite line: E = λ/2πε₀r
- Infinite plane: E = σ/2ε₀
Don't memorize these. Derive them once, then trust the sheet.
Electric Potential — the scalar cousin:
V = k q/r (point charge)
V = -∫ E·dl (general)
E = -∇V (the connection)
Add the dipole potential: V = k p·r̂/r². It shows up in radiation, in molecular physics, in quals. Worth the line.
Poisson and Laplace — the heavy machinery:
∇²V = -ρ/ε₀ (Poisson)
∇²V = 0 (Laplace, charge-free regions)
If you're in a course that does separation of variables, add the general solutions in Cartesian, cylindrical, spherical. If not, skip — but write "separation of variables" as a reminder of the method.
Conductors and Capacitors
Conductor rules — bullet these, don't equation them:
- E = 0 inside
- V = constant throughout
- Charge lives on surface
- E⊥ = σ/ε₀ just outside
- Equipotential surfaces
Capacitance:
C = Q/V
C_parallel_plate = ε₀A/d
C_cylindrical = 2πε₀L/ln(b/a)
C_spherical = 4πε₀ab/(b-a)
Energy stored: U = ½CV² = ½Q²/C = ½∫ε₀E²dV. That last form — field energy density u = ½ε₀E² — is the one that generalizes.
Dielectrics — if your course covers them:
D = ε₀E + P = εE
ε = ε₀(1+χₑ) = ε₀κ
Boundary conditions: D⊥ continuous (no free charge), E∥ continuous
The D-field confuses everyone. Write "D handles free charge only" in the margin.
Magnetostatics: The Curl Side
Biot-Savart Law — the Coulomb equivalent for B:
B = μ₀/4π ∫ I dl' × r̂/r²
Add the infinite wire result: B = μ₀I/2πr. And the loop center: B = μ
…B = μ₀I/(2R) for a single circular loop of radius R carrying steady current I. This result is a special case of the more general Biot‑Savart integral and is useful when estimating the field near coils or magnetic dipoles.
Ampère’s Law – integral and differential forms, alongside the symmetry cases where the line integral simplifies:
∮ B·dl = μ₀ I_enc
∇×B = μ₀ J (plus μ₀ε₀ ∂E/∂t for time‑varying fields; in magnetostatics the displacement term vanishes)
Apply the integral form to:
- Long straight wire: ∮ B·dl = B(2πr) → B = μ₀I/(2πr) (inside and outside the wire the same expression holds for r > wire radius).
- Infinite solenoid (n turns per length): B = μ₀nI inside, B ≈ 0 outside.
- Toroid (N turns, mean radius r): B = μ₀NI/(2πr) within the windings, zero elsewhere.
Magnetic Vector Potential – often easier to handle than B directly for complicated current distributions:
A(r) = μ₀/4π ∫ J(r')/|r−r'| dV'
B = ∇×A
In the Coulomb gauge (∇·A = 0) the potential satisfies Poisson’s equation ∇²A = −μ₀J, mirroring the electrostatic case. For a thin wire, A reduces to a line integral analogous to the Biot‑Savart expression It's one of those things that adds up..
Magnetic Dipole Moment – a compact description of localized current loops:
m = I a (vector magnitude I times area a, direction given by right‑hand rule)
Far‑field approximations:
A(r) ≈ μ₀/4π (m×r̂)/r²
B(r) ≈ μ₀/4π [3(m·r̂)r̂ − m]/r³
These expressions appear in torque τ = m×B, potential energy U = −m·B, and in the interaction of atoms with external fields.
Magnetostatic Boundary Conditions – at an interface between two media (with permeabilities μ₁, μ₂) and surface current K:
B₁⊥ = B₂⊥ (normal component continuous)
H₁∥ − H₂∥ = K × n̂ (tangential H jumps by surface current)
where H = B/μ and n̂ is the unit normal pointing from medium 1 to 2. In the absence of free surface current, H∥ is continuous Took long enough..
Energy in Magnetic Fields – analogous to the electric case:
U = ½∫ B·H dV = ½∫ (B²/μ) dV
u_m = B²/(2μ) (magnetic energy density)
For linear, isotropic media this integrates to the familiar ½LI² for an inductor of inductance L.
Conclusion
This sheet gathers the core vector‑calculus relations that underlie static electric and magnetic fields: Coulomb’s law and superposition, Gauss’s law for E, Ampère’s law (and its integral avatars) for B, the definitions of potential and vector potential, and the key boundary conditions for conductors, dielectrics, and magnetic media. By memorizing the symmetry‑based results (point charge, infinite line, infinite plane, long wire, solenoid, toroid) and the differential forms (∇·E = ρ/ε₀, ∇×B = μ₀J), you can reconstruct any needed expression on the fly. Keep the sheet handy, practice deriving each case once, and let the mathematics guide your intuition rather than the other way around. Good luck on your exams and projects Nothing fancy..
