Foundations — Lenz's law — opposing induced current

Before you can use Lenz's law, you need to be fluent in the words and symbols it quietly assumes. This page builds each one from nothing — plain words, then a picture, then why the topic needs it. Read top to bottom: each block uses only things defined above it.

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1. The magnetic field $\vec{B}$ — invisible arrows filling space

The picture: think of iron filings sprinkled near a magnet — they line up into curved paths. Each filing sits on one arrow of the field.

Why the topic needs it: Lenz's law is entirely about field passing through a loop. Without the field arrows there is nothing to pass through anything.

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2. Into-the-page and out-of-the-page — drawing 3D arrows on flat paper

The field often points through your sheet of paper. We need symbols for that.

Why the topic needs it: the sliding-rod example and every coil diagram show a field piercing the page. You must instantly read these two symbols or the diagrams are gibberish.

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3. Area and the loop's face — $A$ and $\vec{A}$

The picture: a hoop lying flat on a table — its area-arrow points straight up out of the hoop.

Why the topic needs it: "how much field goes through" depends on both the field and how the loop is tilted relative to it. The area-vector is how we bookkeep the tilt.

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4. The angle $\theta$ — how tilted the loop is

The picture: hold a hoop in a rainfall. Face the rain and you catch the most (small angle). Turn the hoop sideways and the rain slips past, catching nothing (right angle).

Why the topic needs it: it controls how much field is "caught," through the $\cos \theta$ factor next.

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5. $\cos \theta$ — the "how much faces the field" dial

WHY cosine and not some other tool? We need a number that is 1 when the loop faces the field head-on ($\theta =0$) and 0 when the loop is edge-on ($\theta =90^{\circ }$), smoothly in between. The cosine function does exactly this: $\cos 0^{\circ }=1$, $\cos 90^{\circ }=0$. It is the natural "fraction that faces you" dial.

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6. Magnetic flux ${\Phi }_B$ — the "how much field is caught" number

Now we can assemble the star of the whole topic. See Magnetic flux for the full story.

The picture: count how many field-arrows thread through the hoop. More threads = more flux.

Why the topic needs it: Lenz's law is a rule about the change in ${\Phi }_B$. Everything else on this page exists to let you compute this one number.

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7. The change $\frac{d{\Phi }_B}{dt}$ — the rate of change tool

WHY do we need a rate of change at all? Lenz's law never cares whether the flux is big or small — only whether it is changing, and how fast. A parked magnet next to a coil, no matter how strong, induces nothing. So we need a tool that measures speed of change.

The picture: a speedometer, but instead of measuring how fast your position changes, it measures how fast the caught field changes.

Why the topic needs it: this rate is what drives the induced push (the EMF), via Faraday's law of induction. No change → no rate → no induction.

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8. EMF $\epsilon$ — the electrical "push" that appears

Why the topic needs it: Faraday's law says $\epsilon =-\frac{d{\Phi }_B}{dt}$. Lenz's law is the meaning of the minus sign — it tells you which direction this push shoves the current. See Motional EMF and sliding rod.

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9. Induced current $I$ and the right-hand rule

The tool that links current to field — the right-hand rule. See Right-hand rule.

Why the topic needs it: Lenz's law first tells you which field the loop must make to oppose the change. The right-hand rule converts that answer into the current direction — the thing you actually report.

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10. The minus sign and conservation of energy

Why the topic needs it: this is the entire content of Lenz's law compressed into one symbol. Understanding it is the goal of the parent note.

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Prerequisite map

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Equipment checklist

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Connections

• Parent: Lenz's law — this page builds every symbol it assumes.
• Magnetic flux — the number ${\Phi }_B$ defined here in full.
• Faraday's law of induction — links the rate of flux change to the EMF.
• Right-hand rule — turns field direction into current direction.
• Motional EMF and sliding rod — where these symbols become a real calculation.
• Conservation of energy — the meaning of the minus sign.