1.5: The Force Density and Torque Density in Matter

The presence of an electric field, \(\vec E\), and a magnetic field,\(\vec B\), in matter results in a force density if the matter is charged and in a torque density if the matter carries electric and magnetic dipole densities. In addition, if the electric field varies in space (the usual case) then a force density is created that is proportional to the electric dipole density and to the electric field gradients. Similarly, if the magnetic field varies in space then a force density is exerted on the matter that is proportional to the magnetic dipole density and to the magnetic field gradients. These force and torque densities are stated below; their proof is left for the problem sets.

1.5.1 The Force Density in Charged and Polarized Matter.

There is a force density that is the direct analogue of Equation (1.1.8), the force acting on a charged particle moving with the velocity \(\vec v\) in electric and magnetic fields, ie

\[ \overrightarrow{\mathrm{f}}=q(\overrightarrow{\mathrm{E}}+[\overrightarrow{\mathrm{v}} \times \overrightarrow{\mathrm{B}}]).\nonumber \]

If this force acting on each charged particle is averaged in time over periods longer than characteristic atomic or molecular orbital times and summed over the particles contained in a volume, ∆V , where ∆V is large compared with atomic or molecular dimensions, then one can divide this total averaged force by ∆V to obtain the force density

\[\overrightarrow{\mathbf{F}}=\rho_{f} \overrightarrow{\mathbf{E}}+\left(\overrightarrow{\mathbf{J}}_{f} \times \overrightarrow{\mathbf{B}}\right) \quad \text { Newtons } / m^{3}.\]

If the electric field in matter varies from place to place there is generated a force density proportional to the dipole moment per unit volume, \(\vec P\), given by

\[\overrightarrow{\mathrm{F}}_{E}=\left(\overrightarrow{\mathrm{P}} \cdot \nabla E_{x}\right) \hat{\mathbf{u}}_{x}+\left(\overrightarrow{\mathrm{P}} \cdot \nabla E_{y}\right) \hat{\mathbf{u}}_{y}+\left(\overrightarrow{\mathrm{P}} \cdot \nabla E_{z}\right) \hat{\mathbf{u}}_{z} \quad \text { Newtons } / m^{3}.\]

In addition, if the magnetic field, \(\vec B\), varies from place to place there will be generated a force density proportional to the magnetic dipole density, \(\vec M\), given by

\[\overrightarrow{\mathrm{F}}_{B}=\left(\overrightarrow{\mathrm{M}} \cdot \nabla B_{x}\right) \hat{\mathbf{u}}_{x}+\left(\overrightarrow{\mathrm{M}} \cdot \nabla B_{y}\right) \hat{\mathbf{u}}_{y}+\left(\overrightarrow{\mathrm{M}} \cdot \nabla B_{z}\right) \hat{\mathbf{u}}_{z} \quad \text { Newtons } / m^{3}.\]

The nabla operator denotes the operation of calculating the gradient of a scalar function \(\phi(\overrightarrow{\mathrm{r}})\). In cartesian co-ordinates

\[\nabla \phi=\frac{\partial \phi}{\partial x} \hat{\mathbf{u}}_{x}+\frac{\partial \phi}{\partial y} \hat{\mathbf{u}}_{y}+\frac{\partial \phi}{\partial z} \hat{\mathbf{u}}_{z}.\nonumber\]

1.5.2 The Torque Densities in Polarized Matter.

It can be shown that an electric field exerts a torque on polarized matter. The torque density is given by

\[\overrightarrow{\mathrm{T}}_{E}=\overrightarrow{\mathrm{P}} \times \overrightarrow{\mathrm{E}} \quad \text { Newtons } / \mathrm{m}^{2}.\]

The magnetic field also exerts a torque on magnetized matter. This torque density is given by

\[\overrightarrow{\mathrm{T}}_{B}=\overrightarrow{\mathrm{M}} \times \overrightarrow{\mathrm{B}} \quad \text { Newtons } / \mathrm{m}^{2}.\]