# 5: The Magnetic Field

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The ancient Chinese knew that the iron oxide magnetite ($$\ce{FesO4}$$) attracted small pieces of iron. The first application of this effect was the navigation compass, which was not developed until the thirteenth century. No major advances were made again until the early nineteenth century when precise experiments discovered the properties of the magnetic field.

• 5.1: Forces on Moving Charges
It was well known that magnets exert forces on each other, but in 1820 Oersted discovered that a magnet placed near a current carrying wire will align itself perpendicular to the wire.
• 5.2: Magnetic Field Due to Currents
Once it was demonstrated that electric currents exert forces on magnets, Ampere immediately showed that electric currents also exert forces on each other and that a magnet could be replaced by an equivalent current with the same result. Now magnetic fields could be turned on and off at will with their strength easily controlled.
• 5.3: Divergence and Curl of the Magnetic Field
Because of our success in examining various vector operations on the electric field, it is worthwhile to perform similar operations on the magnetic field.
• 5.4: The Vector Potential
Since the divergence of the magnetic field is zero, we may write the magnetic field as the curl of a vector,
• 5.5: Magnetization
Our development thus far has been restricted to magnetic fields in free space arising from imposed current distributions. Just as small charge displacements in dielectric materials contributed to the electric field, atomic motions constitute microscopic currents, which also contribute to the magnetic field. There is a direct analogy between polarization and magnetization, so our development will parallel that of Section 3-1.
• 5.6: Boundary Conditions
At interfacial boundaries separating materials of differing properties, the magnetic fields on either side of the boundary must obey certain conditions. The procedure is to use the integral form of the field laws for differential sized contours, surfaces, and volumes in the same way as was performed for electric fields in Section 3-3.
• 5.7: Magnetic Field Boundary Value Problems
A line current I of infinite extent in the z direction is a distance d above a plane that is either perfectly conducting or infinitely permeable, as shown in Figure 5-24. For both cases
• 5.8: Magnetic Fields and Forces
A magnetizable medium carrying a free current $$\textbf{J}_{f}$$ is placed within a magnetic field $$\textbf{B}$$, which is a function of position. In addition to the Lorentz force, the medium feels the forces on all its magnetic dipoles. Focus attention on the rectangular magnetic dipole shown in Figure 5-26. The force on each current carrying leg is
• 5.9: Problems

Thumbnail: Magnetic fields can be visualized with iron filings, that align along the magnetic field direction. Here the magnetic field of a homogeneously magnetized cylindrical bar magnet was accurately computed, and the field is shown with simulated randomly placed iron filings. The density of filings is also proportional to the field strength. The field is strongest around the magnetic poles. (CC BY-SA 4.0; Geek3 via Wikipedia)

This page titled 5: The Magnetic Field is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Markus Zahn (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.