# Chapter 5 Magnetostatics

# Objectives

Upon learning the material presented in this chapter, you should be able to:

1. Calculate the magnetic force on a current-carrying wire placed in a magnetic field and the torque exerted on a current loop.

2. Apply the Biot–Savart law to calculate the magnetic field due to current distribution.

3. Apply Ampère’s law to configurations with appropriate symmetry.

4. Explain magnetic hysteresis in ferromagnetic materials.

5. Calculate the inductance of a solenoid, a coaxial transmission line, or other configurations.

6. Relate the magnetic energy stored in a region to the magnetic field distribution in that region.

# Overview

This chapter on magnetostatics parallels the preceding one on electrostatics. Stationary charges produce static electric fields, and steady (i.e., non–time-varying) currents produce static magnetic fields. When ∂/∂t=0, the magnetic fields in a medium with magnetic permeability μ are governed by the second pair of Maxwell’s equations (Eqs. (4.3a,b)):

where J is the current density. The magnetic flux density B and the magnetic field intensity H are related by

B=μ H.(5.2)

When examining electric fields in a dielectric medium in Chapter 4, we noted that the relation D=ε E is valid only when the medium is linear and isotropic. These properties, which hold true for most materials, allow us to treat the permittivity ε as a constant, scalar quantity that is independent of both the magnitude and the direction of E. A similar statement applies to the relation given by Eq. (5.2). With the exception of ferromagnetic materials, for which the relationship between B and H is nonlinear, most materials are characterized by constant permeabilities.

The objectives of this chapter are to develop an understanding of the relationship between steady currents and the magnetic flux B and field H due to various types of current distributions and in various types of media and to introduce a number of related quantities, such as the magnetic vector potential A, the magnetic energy density w m, and the inductance of a conducting structure, L. The parallelism that exists between these magnetostatic quantities and their electrostatic counterparts is elucidated in Table 5-1.

# Table 5-1 Attributes of electrostatics and magnetostatics.

Table 5 1 Full Alternative Text