Foundations — Faraday's law — EMF = −dΦ - dt

This is the toolbox page for Faraday's law. The parent note throws around $\vec{B}$, $\hat{n}$, $\cos \theta$, dot products, derivatives, and the letters $\mathcal{E}$ and ${\Phi }_B$ as if you already own them. Here we earn each one, in an order where every symbol leans only on the ones before it.

1. Arrows that carry a direction: vectors

Why do we need arrows at all? Because a magnetic field is not just "how strong" — it also has a "which way." A field pointing straight through a loop does something completely different from a field sliding sideways past it. A single number cannot tell those apart; an arrow can.

2. The magnetic field arrow: $\vec{B}$

We track $\vec{B}$ because it is the cause in this whole story. Everything Faraday's law does, it does by watching $\vec{B}$ (and the loop) change.

3. The loop and its area: $A$ and the normal $\hat{n}$

But a flat surface sitting in 3-D space also has a facing direction — which way is it "looking"? That direction is captured by a special little arrow.

Why invent $\hat{n}$? Because to ask "is the field poking through the loop or sliding along it?" we need something to compare $\vec{B}$ against. The normal is the honest representative of "through-ness": if $\vec{B}$ lines up with $\hat{n}$, the field pokes straight through; if $\vec{B}$ is at right angles to $\hat{n}$, the field skims the surface and threads nothing.

4. The angle $\theta$ and why $\cos$ appears

Now we need a tool that turns this angle into a number saying how much of the field goes straight through. That tool is the cosine.

Why cosine and not something else? We are asking a projection question: "of this tilted field, how much lies along the through-direction $\hat{n}$?" Projection onto a direction is exactly what cosine computes. Sine would answer the opposite question ("how much lies sideways"), which threads nothing — so cosine is the right tool.

Watch every case so nothing surprises you later:

That negative case matters: it is how the loop can tell "field entering the front" from "field entering the back," which later becomes the direction of the induced current.

5. The dot product: $\vec{B}\cdot d\vec{A}$

The parent writes flux as ${\int }_S\vec{B}\cdot d\vec{A}$. That dot is the dot product, and it is just the projection idea from §4 packaged for two arrows.

Here $\vec{A}$ means "the area vector": an arrow whose length is the area $A$ and whose direction is the normal $\hat{n}$. So $\vec{B}\cdot \vec{A}$ is literally "field strength $\times$ area $\times$ (how aligned they are)" — exactly what we want flux to mean. The ${\int }_S$ (an integral) just means "add up these little contributions over the whole surface"; for a flat loop in a uniform field it collapses to one clean product, $BA\cos \theta$.

6. Magnetic flux: ${\Phi }_B$

Now we can assemble the star quantity.

Every symbol in $BA\cos \theta$ is now something you built: $B$ the field's strength (§2), $A$ the loop's area (§3), $\cos \theta$ the alignment fraction (§4). This is the quantity Faraday's law watches. Deep-dive: Magnetic Flux.

7. Change over time: the derivative $\frac{d{\Phi }_B}{dt}$

Faraday's law is not about flux — it is about flux changing. So we need a symbol for "rate of change."

Why a derivative and not just a difference? A magnet can be pushed in smoothly, speeding up and slowing down. We want the instantaneous rate at every moment, not just "how much changed in total." The derivative is the tool that gives the exact slope at a single instant. The little $d$'s mean "an infinitely tiny change" — so $d{\Phi }_B/dt$ is a tiny flux change divided by the tiny time it took.

8. The output: EMF, written $\mathcal{E}$

Putting the pieces together, the parent's law $\mathcal{E}=-\frac{d{\Phi }_B}{dt}$ now reads in plain words: the push-per-charge the loop makes equals how fast the spaghetti-count is changing, with a minus sign. The minus sign is Lenz's law — the loop pushes back to resist the change — and it gets its own deep dive at Lenz's Law.

9. Prerequisite map

Equipment checklist

Test yourself — you are ready for the parent note when you can answer each without peeking.