Applied Physics » Electromagnetism » Magnetostatics
Description:
Magnetostatics is a subfield of electromagnetism that deals with magnetic fields in systems where the currents are steady (time-invariant). In essence, it is the study of magnetic fields in conditions such that they do not change with time. These static fields are produced by steady currents, and the theoretical foundation for understanding them is laid by steady-state solutions to Maxwell’s equations, specifically in the absence of time-varying electric fields.
In magnetostatics, the primary concern is with the magnetic field B and its relationship to the steady current density J. The central equations governing magnetostatics are derived from two of Maxwell’s equations, specifically:
Ampère’s Law (with Maxwell’s correction term omitted):
\[
\nabla \times \mathbf{B} = \mu_0 \mathbf{J}
\]
where \(\mathbf{B}\) is the magnetic flux density (or magnetic field), \(\mu_0\) is the permeability of free space, and \(\mathbf{J}\) is the current density.Gauss’s Law for Magnetism:
\[
\nabla \cdot \mathbf{B} = 0
\]
which states that there are no “magnetic charges”; this implies the magnetic field lines are continuous and form closed loops.
One of the main tools in magnetostatics is the magnetic vector potential A, which is defined such that:
\[
\mathbf{B} = \nabla \times \mathbf{A}
\]
This potential is useful because it simplifies the process of solving Ampère’s Law, especially in complex geometries.
Moreover, the Biot-Savart Law is a fundamental principle in magnetostatics used to calculate the magnetic field produced at a point due to a steady current. The law is given by:
\[
\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int \frac{\mathbf{J}(\mathbf{r}‘) \times (\mathbf{r} - \mathbf{r’})}{|\mathbf{r} - \mathbf{r’}|^3} d^3\mathbf{r’}
\]
where \(\mathbf{r}\) is the position where the magnetic field is being calculated, and \(\mathbf{r’}\) is the position of the current element.
Applications of magnetostatics can be seen in the design of electromagnets, magnetic circuit design, and understanding the behavior of permanent magnets. The principles are crucial for technologies ranging from electric motors and transformers to magnetic storage devices.
Understanding magnetostatics is fundamental for students and professionals dealing with various applications in applied physics and electrical engineering, providing the core knowledge needed to manipulate and design systems that utilize steady magnetic fields.