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.

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.

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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

$d\vec{B} = \frac{\mu_0}{4\pi} \frac{I \, d\vec{l} \times \hat{r}}{r^2}$

This is the fundamental field equation. It describes the tiny contribution $d\vec{B}$ to the magnetic field produced by an infinitesimal current element $I \, d\vec{l}$ at a distance $r$.

• 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 $\hat{r}$, and your thumb points along $d\vec{B}$
• To find the total field from an entire wire or loop, you integrate $d\vec{B}$ over the full current path

Magnetic Field of a Long Straight Wire

$B = \frac{\mu_0 I}{2\pi r}$

The field decreases as you move farther from the wire (inversely proportional to distance $r$). Field lines form concentric circles around the wire. Use the right-hand rule with your thumb along the current direction; your fingers curl in the direction of $\vec{B}$.

This formula is your starting point for problems involving parallel wires or wire interactions.

Magnetic Field at the Center of a Circular Loop

$B = \frac{\mu_0 I}{2R}$

The field at the center depends inversely on the loop radius $R$. Its direction is perpendicular to the plane of the loop: curl your right-hand fingers along the current, and your thumb points along $\vec{B}$.

For $N$ identical loops stacked together (a flat coil), the field multiplies: $B = \frac{\mu_0 N I}{2R}$. This is the building block for understanding solenoids and electromagnets.

Magnetic Field of a Solenoid

$B = \mu_0 n I$

Here $n$ is the number of turns per unit length ($n = N/L$). The field inside a long solenoid is remarkably uniform, with parallel field lines running along the axis. The field outside is approximately zero.

This is a high-symmetry case that Ampère's Law handles cleanly. Expect it on exams.

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Ampère's Law: The Symmetry Shortcut

When current distributions have high symmetry, Ampère's Law provides an elegant way to calculate the magnetic field without doing a full Biot-Savart integration.

Ampère's Law

$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$

This relates the line integral of $\vec{B}$ around a closed loop (called an Amperian loop) to the total current $I_{enc}$ enclosed by that loop.

The catch: it's only practical when $\vec{B}$ is constant in magnitude along your chosen loop, so the dot product simplifies. This happens with cylindrical symmetry (long straight wires), solenoidal symmetry (solenoids), and toroidal symmetry (toroids).

Think of it as analogous to Gauss's Law for electric fields. The strategy is the same: choose your loop wisely so the math collapses.

Steps for applying Ampère's Law:

1. Identify the symmetry of the current distribution
2. Choose an Amperian loop where $\vec{B}$ is either constant and parallel to $d\vec{l}$, or perpendicular to $d\vec{l}$ (contributing zero)
3. Evaluate $\oint \vec{B} \cdot d\vec{l}$, which typically simplifies to $B \times (\text{loop length})$
4. Determine $I_{enc}$, the total current passing through the loop
5. Set them equal and solve for $B$

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Force Interactions: Fields Acting on Charges and Currents

Magnetic fields don't just exist passively. They exert forces on moving charges and current-carrying conductors.

Lorentz Force (Force on a Moving Charge)

$\vec{F} = q(\vec{v} \times \vec{B})$

The force is perpendicular to both the velocity $\vec{v}$ and the field $\vec{B}$. Its magnitude is $F = qvB\sin\theta$, where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$.

• No work done: because the force is always perpendicular to the velocity, magnetic forces change a charge's direction but never its speed (and therefore never its kinetic energy)
• Circular motion results when $\vec{v} \perp \vec{B}$. Setting the magnetic force equal to centripetal force gives the radius of the circular orbit: $r = \frac{mv}{qB}$. This principle underlies mass spectrometers and cyclotrons.

Magnetic Force on a Current-Carrying Wire

$\vec{F} = I(\vec{L} \times \vec{B})$

This gives the force on a straight wire segment of length $L$ carrying current $I$ in an external field $\vec{B}$. The vector $\vec{L}$ points in the direction of current flow. The magnitude is $F = ILB\sin\theta$.

Use the right-hand rule: point fingers along $\vec{L}$ (current direction), curl toward $\vec{B}$, and your thumb gives $\vec{F}$. This force is what makes electric motors rotate.

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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

$\Phi_B = \vec{B} \cdot \vec{A} = BA\cos\theta$

Magnetic flux measures how much magnetic field passes through a given surface. The angle $\theta$ is between $\vec{B}$ and the area normal (the vector perpendicular to the surface). Flux is maximized when the field is perpendicular to the surface ($\theta = 0°$, so $\cos\theta = 1$) and zero when the field runs parallel to the surface ($\theta = 90°$).

You need to understand flux before Faraday's Law makes sense, because flux is the quantity whose change drives induction.

Faraday's Law of Induction

$\mathcal{E} = -N\frac{d\Phi_B}{dt}$

The induced emf in a coil of $N$ turns equals the negative of the rate of change of magnetic flux through the coil. For a single loop, $N = 1$.

There are three ways to change flux (and exam problems test all three):

1. Change $B$: increase or decrease the field strength
2. Change $A$: expand, shrink, or deform the loop area (e.g., a sliding rail on a track)
3. Change $\theta$: rotate the loop relative to the field (this is how generators work)

A rotating coil in a constant magnetic field continuously changes $\theta$, producing a sinusoidal AC voltage.

Lenz's Law

The induced current flows in whatever direction creates a magnetic field that opposes the change in flux that caused it. If flux through a loop is increasing, the induced current produces a field opposing the increase. If flux is decreasing, the induced current tries to maintain it.

This is conservation of energy in disguise. If induced currents aided the flux change instead of opposing it, you'd get a runaway process that creates energy from nothing.

Mathematically, Lenz's Law is encoded in the negative sign in Faraday's Law. On problems, use Faraday's Law to find the emf magnitude, then use Lenz's Law to determine which way the current flows.

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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?