❓How do the static and source-free conditions simplify the expression for the total force on the ele

How do the static and source-free conditions simplify the expression for the total force on the electromagnetic field?

The static ($\partial/\partial t = 0$) and source-free ($\rho = 0, \mathbf{J} = 0$) conditions simplify the total electromagnetic force on the field ($\mathbf{F}_{\text{field}}$) by ensuring that the force density is zero everywhere within the volume.

The simplification can be understood through two equivalent perspectives: the Lorentz force density and the Maxwell stress tensor.

The total force on the electromagnetic field ($\mathbf{F}{\text{field}}$) is the negative of the total force on the matter ($\mathbf{F}{\text{matter}}$) inside the volume $V$, where $\mathbf{f}_{\text{matter}}$ is the Lorentz force density:

$\mathbf{F}{\text{field}} = - \mathbf{F}{\text{matter}} = - \int_V \mathbf{f}_{\text{matter}} \, d\tau = - \int_V (\rho\mathbf{E} + \mathbf{J} \times \mathbf{B}) \, d\tau$

The source-free condition ($\rho = 0, \mathbf{J} = 0$) provides the primary and most direct simplification:

• Since there are no charges ($\rho=0$) and no currents ($\mathbf{J}=0$) inside the volume, the Lorentz force density is zero everywhere: $\mathbf{f}_{\text{matter}} = (0)\mathbf{E} + (\mathbf{0}) \times \mathbf{B} = \mathbf{0}$.

• Therefore, the integral for the total force on the field is immediately zero:

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

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How do the static and source-free conditions simplify the expression for the total force on the electromagnetic field? | FAQsviadean on Notion