Magnetic Field Formulas

Why This Matters

Magnetic field formulas are the backbone of electromagnetism—one of the most heavily tested topics in Physics II. You're being tested on your ability to connect current flow to field generation, field geometry to symmetry, and changing flux to induced voltage. These aren't isolated equations; they form a coherent story about how moving charges create fields, how those fields exert forces, and how changing fields generate new currents. Understanding this chain of cause and effect is what separates students who ace FRQs from those who struggle to apply formulas.

The key principles you'll need to master include field generation (how currents produce magnetic fields), force interactions (how fields act on charges and wires), and electromagnetic induction (how changing fields create voltage). Don't just memorize formulas—know what physical situation each formula describes and when to apply it. If you can identify the symmetry of a problem and connect it to the right law, you'll navigate even the trickiest exam questions with confidence.

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Field Generation: How Currents Create Magnetic Fields

Moving charges produce magnetic fields, and the geometry of the current flow determines the field's shape and strength. These formulas let you calculate the magnetic field produced by different current configurations.

Biot-Savart Law

• Fundamental field equation—describes the magnetic field $d\vec{B}$ produced by an infinitesimal current element $Id\vec{l}$
• Inverse-square dependence: field strength falls off as $1/r^2$ from the current element, similar to Coulomb's law for electric fields
• Right-hand rule determines direction—curl fingers from $d\vec{l}$ toward $\hat{r}$, thumb points along $d\vec{B}$

Magnetic Field of a Long Straight Wire

• Formula: $B = \frac{\mu_0 I}{2\pi r}$—field decreases linearly with distance $r$ from the wire
• Circular field lines wrap around the wire in concentric circles; use right-hand rule with thumb along current
• Practical baseline—this is often your starting point for problems involving parallel wires or wire interactions

Magnetic Field at the Center of a Circular Loop

• Formula: $B = \frac{\mu_0 I}{2R}$—field at center depends inversely on loop radius $R$
• Direction is perpendicular to the loop plane; right-hand rule with fingers curling along current gives thumb pointing along $\vec{B}$
• Building block for understanding solenoids and electromagnets—multiple loops amplify this effect

Magnetic Field of a Solenoid

• Formula: $B = \mu_0 n I$—uniform field inside depends on turns per unit length $n$ and current $I$
• Uniform interior field with parallel field lines; field outside is approximately zero
• High-symmetry case—Ampère's Law makes this derivation straightforward; expect this on exams

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Ampère's Law: The Symmetry Shortcut

When current distributions have high symmetry, Ampère's Law provides an elegant path to calculating the magnetic field without integration.

Ampère's Law

• Circulation equation: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$—relates the line integral of $\vec{B}$ around a closed loop to enclosed current
• Symmetry requirement—only useful when $\vec{B}$ is constant along your chosen Amperian loop (cylindrical, toroidal, or solenoidal symmetry)
• Analogous to Gauss's Law for electric fields; choose your loop wisely to simplify calculations

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Force Interactions: Fields Acting on Charges and Currents

Magnetic fields don't just exist—they exert forces on moving charges and current-carrying conductors. These formulas describe how fields and charges interact.

Lorentz Force (Force on a Moving Charge)

• Formula: $\vec{F} = q(\vec{v} \times \vec{B})$—force is perpendicular to both velocity and field
• No work done—because force is always perpendicular to motion, magnetic forces change direction but not speed
• Circular motion results when $\vec{v} \perp \vec{B}$; this principle underlies mass spectrometers and cyclotrons

Magnetic Force on a Current-Carrying Wire

• Formula: $\vec{F} = I(\vec{L} \times \vec{B})$—force on wire segment of length vector $\vec{L}$ carrying current $I$
• Right-hand rule applies: point fingers along $\vec{L}$ (current direction), curl toward $\vec{B}$, thumb gives $\vec{F}$
• Motor principle—this force is what makes electric motors rotate; expect applications questions

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Electromagnetic Induction: Changing Fields Create Voltage

A changing magnetic environment induces electric effects—this is the principle behind generators, transformers, and wireless charging.

Magnetic Flux

• Definition: $\Phi_B = \vec{B} \cdot \vec{A} = BA\cos\theta$—measures total field passing through a surface
• Angle matters—$\theta$ is between $\vec{B}$ and the surface normal; maximum flux when field is perpendicular to surface
• Gateway concept—you must understand flux before Faraday's Law makes sense; it's the quantity that changes

Faraday's Law of Induction

• Formula: $\mathcal{E} = -\frac{d\Phi_B}{dt}$—induced emf equals negative rate of flux change
• Three ways to change flux: vary $B$, vary $A$, or vary $\theta$—exam problems often test all three
• Generator principle—rotating a coil in a magnetic field continuously changes flux, producing AC voltage

Lenz's Law

• Opposition principle—induced current creates a magnetic field that opposes the change in flux that caused it
• Conservation of energy in disguise—if induced currents aided the change, you'd get energy from nothing
• Negative sign in Faraday's Law encodes Lenz's Law mathematically; use it to determine current direction

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Quick Reference Table

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Self-Check Questions

1. A long straight wire and a solenoid both carry the same current. How does the magnetic field's spatial dependence differ between them, and why does this difference arise from their geometries?

2. You need to calculate the magnetic field around a current-carrying wire. Under what conditions would you choose Ampère's Law over the Biot-Savart Law, and what must be true about the field for Ampère's Law to be useful?

3. Compare the Lorentz force on a single moving charge to the force on a current-carrying wire. How are the formulas related, and why does a magnetic force do no work on a free charge but can do work on a wire?

4. A bar magnet approaches a conducting loop. Using both Faraday's Law and Lenz's Law, explain how to determine (a) whether an emf is induced, (b) its magnitude, and (c) the direction of the induced current.

5. If you wanted to maximize the induced emf in a generator, which variables in Faraday's Law would you optimize, and what practical trade-offs might limit each approach?