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:
$\mathbf{F}_{\text{field}} = - \int_V \mathbf{0} \, d\tau = \mathbf{0}$
The total force on the field is generally expressed in terms of the Maxwell stress tensor ($\mathbf{T}$) and the time derivative of the electromagnetic momentum density ($\mathbf{g}$):
$$ \mathbf{F}_{\text{field}} = \int_V \left(\mathbf{\nabla} \cdot \mathbf{T} - \frac{\partial \mathbf{g}}{\partial t}\right) d\tau $$
The static condition ($\partial/\partial t = 0$) eliminates the rate of change of field momentum:
$$ \frac{\partial \mathbf{g}}{\partial t} = \mathbf{0} $$
This removes the transient force term, simplifying the expression to:
$$ \mathbf{F}_{\text{field}} = \int_V (\mathbf{\nabla} \cdot \mathbf{T}) d\tau $$
The source-free condition ($\rho = 0, \mathbf{J} = 0$) ensures that in this static region, the force density $\mathbf{f}{\text{field}} = \mathbf{\nabla} \cdot \mathbf{T}$ must also be zero (since $\mathbf{f}{\text{field}} = -\mathbf{f}{\text{matter}}$, and $\mathbf{f}{\text{matter}}=0$):
$$ \mathbf{\nabla} \cdot \mathbf{T} = \mathbf{0} $$
This means that while the static condition removes the time-dependent term, the source-free condition ensures that the remaining term ($\mathbf{\nabla} \cdot \mathbf{T}$) is also zero, leading to the same result: $\mathbf{F}_{\text{field}} = \mathbf{0}$. Physically, the static, source-free conditions mean that the field momentum within the volume is constant, and the internal stresses exactly balance out.