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

The total electromagnetic force on the electromagnetic field inside a volume is zero if the volume is source-free and the situation is static.

Here is the explanation based on the principles of electromagnetism, as described in the shared document:

The force density ($\mathbf{f}_{\text{matter}}$) exerted by the electromagnetic field on matter (charges and currents) is given by the Lorentz force density:

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

The total force on the field ($\mathbf{F}_{\text{field}}$) is the negative of the total force on the matter, according to Newton's third law:

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

Applying the Conditions

The problem states two key conditions for the volume $V$:

1. Source-Free: There are no charges or currents inside the volume, meaning the charge density is zero ($\rho = 0$) and the current density is zero ($\mathbf{J} = 0$).

2. Static: The situation is not changing with time, meaning the time derivative is zero ($\partial/\partial t = 0$).

When the source-free condition is applied to the integral for the total force on the field:

$\mathbf{F}_{\text{field}} = - \int{V} (\mathbf{0} \cdot \mathbf{E} + \mathbf{0} \times \mathbf{B}) \, d\tau = - \int_{V} (\mathbf{0}) \, d\tau = \mathbf{0}$

Since the integrand ($\rho \mathbf{E} + \mathbf{J} \times \mathbf{B}$) is zero everywhere inside the volume $V$, the total volume integral is zero.

The static condition ($\partial/\partial t = 0$) is also important in the full derivation using the Maxwell Stress Tensor ($\mathbf{T}$), which shows the total force on the field can be written as:

$\mathbf{F}_{\text{field}} = \int{V} \left(\nabla \cdot \mathbf{T} - \frac{\partial \mathbf{g}}{\partial t}\right) \, d\tau$

where $\mathbf{g}$ is the linear momentum density of the electromagnetic field. The static condition ensures the time-derivative term ($\partial \mathbf{g}/\partial t$) is zero, simplifying the expression. However, for a source-free region, the most direct result is simply that the force density itself is zero.

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Proof and Derivation related to FAQ

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