❓What is the force density of the electromagnetic field on matter (Lorentz force density)?

The Lorentz force density ($\mathbf{f}{\text{matter}}$) is the force per unit volume exerted by an electromagnetic field on matter (charges and currents). It is mathematically defined as: $\mathbf{f}{\text{matter}} = \rho\mathbf{E} + \mathbf{J} \times \mathbf{B}$

This expression combines two distinct contributions to the total force:

This term represents the force exerted by the electric field ($\mathbf{E}$) on any net charge present within the volume.

• $\mathbf{\rho}$ is the charge density (charge per unit volume).

• $\mathbf{E}$ is the electric field vector.

• This component acts parallel or anti-parallel to the electric field lines, pushing static charges.

This term represents the force exerted by the magnetic field ($\mathbf{B}$) on any current (moving charges) within the volume.

• $\mathbf{J}$ is the current density (current per unit area).

• $\mathbf{B}$ is the magnetic field vector.

• This component acts perpendicular to both the current density and the magnetic field, a characteristic of the magnetic Lorentz force.

In summary, the Lorentz force density is a local expression that determines the total force on a piece of matter by adding up the electric push on all the charges ($\rho$) and the magnetic push on all the moving charges ($\mathbf{J}$) in that tiny volume element.

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