❓Why is the total electromagnetic force zero in a source-free static volume?

The total electromagnetic force in a source-free static volume is zero because the Lorentz force density is zero throughout the volume.

Here is the explanation, directly following the principles provided:

Lorentz Force and Force on the Field

1. Force on Matter ($\mathbf{f}_{\text{matter}}$): The force per unit volume exerted by the electromagnetic field on any matter (charges and currents) inside a volume ($V$) is given by the Lorentz force density:

   $\mathbf{f}_{\text{matter}} = \rho\mathbf{E} + \mathbf{J} \times \mathbf{B}$

   where $\rho$ is the charge density, $\mathbf{J}$ is the current density, $\mathbf{E}$ is the electric field, and $\mathbf{B}$ is the magnetic field.

2. Force on the Field ($\mathbf{F}{\text{field}}$): By Newton's third law, the total force exerted by the matter on the field ($\mathbf{F}_{\text{field}}$) is the negative of the total force exerted by the field on the matter ($\mathbf{F}_{\text{matter}}$):

   $\mathbf{F}_{\text{field}} = - \mathbf{F}{\text{matter}} = - \int_V \mathbf{f}_{\text{matter}} \, d\tau$

Applying Source-Free Conditions

The problem defines a source-free volume, which means:

• The charge density is zero: $\rho = 0$.

• The current density is zero: $\mathbf{J} = 0$.

Substituting these conditions into the Lorentz force density equation:

$\mathbf{f}_{\text{matter}} = (0)\mathbf{E} + (\mathbf{0}) \times \mathbf{B} = \mathbf{0}$

Since the force density on matter is zero at every point inside the volume, the total force on the matter is zero, and therefore, the total force on the field is also zero:

$\mathbf{F}{\text{field}} = - \int_V \mathbf{f}{\text{matter}} \, d\tau = - \int_V \mathbf{0} \, d\tau = \mathbf{0}$

In short, a source-free region contains nothing for the field to push on, so the net force exchange between the field and matter within that volume is zero. The static condition ($\partial/\partial t = 0$) ensures there are no time-varying field momentum effects that would contribute to a non-zero force integral.

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Proof and Derivation related to FAQ

Why is the total electromagnetic force zero in a source-free static volume? | FAQsviadean on Notion