When you apply an external magnetic field to a material, two things are happening at once. First, you have the external field itself, called the magnetic field strength, represented by $H$. Second, the material reacts to this field by aligning its own internal magnetic dipoles. This internal response is called magnetization, or $M$.

The total magnetic field inside the material, known as magnetic flux density or $B$, is the sum of both the external field and the material's internal response. This relationship is one of the cornerstones of magnetism.

Different materials respond to an $H$ field with varying enthusiasm. Some barely react, while others align their internal dipoles very strongly. We quantify this responsiveness using a property called magnetic susceptibility, represented by the Greek letter chi, $\chi$.

A high susceptibility means a material magnetizes easily. We can substitute this back into our first equation to see how B and H are related through the material's properties.

$B = \mu_0 (H + \chi H)$ $B = \mu_0 (1 + \chi) H$

The term $(1 + \chi)$ is so useful it gets its own name: relative permeability ($μ_r$). It compares a material's permeability to that of a vacuum.

In practical magnetic design, you'll encounter two different ways of thinking about these fields. Neither is wrong; they're just different conventions for different problems.

The Engineering Approach: Engineers designing motors or sensors often treat the $H$ field as the independent variable—the 'drive' they create with electric currents. The $B$ field is the resulting response from the magnetic materials they use. Their goal is to maximize $B$ for a given $H$.

The Physics Approach: Physicists often consider the $B$ field to be more fundamental. It's the field that actually exerts forces on moving charges (the Lorentz force). In this view, $H$ is a calculated quantity that accounts for how materials contribute to the total field.

This distinction matters when you're working with materials like permanent magnets versus soft magnetic cores for electromagnets. A permanent magnet has a large, built-in $M$ and a low relative permeability ($μ_r ≈ 1.05$). It doesn't respond much to an external $H$ field because it's already maxed out. In contrast, a soft iron core has a tiny initial $M$ but a huge $μ_r$ (often > 5000), meaning even a small $H$ field can induce a massive $B$ field.

Understanding these nuances is key. It allows you to select the right materials and predict their performance, bridging the gap between abstract physics and the design of powerful, efficient magnetic devices.