Summary

General Principles

At its most fundamental level, magnetism is an interaction between moving charges. The magnetic field of one moving charge exerts a force on another moving charge.

The figure consists of two parts that illustrates the magnetic force for a positive point charge in a uniform magnetic field.

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Magnetic Fields

The Biot-Savart law for a moving point charge

\vec{B} = \frac{μ_{0}}{4 π} \frac{q \vec{v} \times \hat{r}}{r^{2}}

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Magnetic field of a current

Model Model wires as simple shapes.

Visualize Divide the wire into short segments.

Solve Use superposition:

Find the field of each segment $Δ s$ .
Find $\vec{B}$ by summing the fields of all $Δ s$ , usually as an integral.

An alternative method for fields with a high degree of symmetry is Ampère’s law:

\oint \vec{B} \cdot d \vec{s} = μ_{0} I_{through}

where $I_{through}$ is the current through the area bounded by the integration path.

Magnetic Forces

The magnetic force on a moving charge is

\vec{F} = q \vec{v} \times \vec{B}

The force is perpendicular to $\vec{v}$ and $\vec{B}$ .

The figure illustrates the magnetic force for moving a positive point charge in a uniform magnetic field.

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The magnetic force on a current-carrying wire is

\vec{F} = I \vec{l} \times \vec{B}

$\vec{F} = \vec{0}$ for a charge or current moving parallel to $\vec{B}$ .

The figure illustrates the magnetic force on a current carrying wire.

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The magnetic torque on a magnetic dipole is

\vec{τ} = \vec{m} \times \vec{B}

The figure illustrates the magnetic torque on a magnetic dipole.

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Applications

Wire

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Loop

The figure illustrates the magnetic field on a flat coil oriented horizontally.

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B_{center} = \frac{μ_{0}}{2}_{} \frac{N I}{R}

Solenoid

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Flat magnet

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Right-hand rule

Point your right thumb in the direction of I. Your fingers curl in the direction of $\vec{B}$ . For a dipole, $\vec{B}$ emerges from the side that is the north pole.

Charged-particle motion

No force if $\vec{v}$ is parallel to $\vec{B}$

The figure illustrates the motion of a charged particle.

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Circular motion at the cyclotron frequency $f_{cyc} = q B / 2 π m$ if $\vec{v}$ is perpendicular to $\vec{B}$

The figure illustrates the circular motion of a charge.

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Parallel wires and current loops

Parallel currents attract.

The figure consists of two parts that illustrate magnetic force acting on parallel lines.

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Opposite currents repel.

The figure consists of two parts that illustrate two alternate ways to view magnetic forces on current loops.

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