# 5.7: The Maxwell Stress Tensor

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In analogy with the electrostatic case, the forces due to the magnetic field acting on the current distribution in a body can be obtained from a magnetic Maxwell stress tensor, see J.A.Stratton, Electromagnetic Theory, section 2.5, (McGraw-Hill, N.Y., 1941). If the magnetic materials in the system are linear so that \(\vec B\) is proportional to \(\vec H\), it can be shown that there exists a vector \(\vec T\)_{M} associated with the elements of the stress tensor such that the surface integral of \(\vec T\)_{M }over a closed surface S gives the net force acting on the material in the volume V enclosed by the surface S: it is assumed that the surface S is contained entirely within a fluid that can support no shearing stresses. The magnetic force acting on the material within the volume V can be calculated from

\[\overrightarrow{\mathrm{F}}_{M}=\int \int_{S} \overrightarrow{\mathrm{T}}_{M} \cdot \overrightarrow{\mathrm{d} \mathrm{S}}, \label{5.75}\]

where the magnitude of the Maxwell stress vector for a linear, isotropic material, is

\[\left|\overrightarrow{\mathrm{T}}_{M}\right|=\frac{\overrightarrow{\mathrm{B}} \cdot \overrightarrow{\mathrm{H}}}{2}, \label{5.76}\]

and its direction is given by the construction shown in Figure (5.6.14). The stress vector \(\vec T\)_{M }is turned away from the surface normal through an angle that is twice the angle that the magnetic field \(\vec B\) (or \(\vec H\)) makes with the surface normal. When \(\vec B\) lies along the surface normal the magnetic force is a tension, but when the field \(\vec B\) lies in the surface the magnetic force is a pressure.