The total electromagnetic force on the matter inside a volume is determined by the volume integral of the Lorentz force density $f=\rho E+J \times B$. The problem specifies a source-free volume $V$, meaning it contains neither free charge ( $\rho=0$ ) nor free current ( $J=0$ ). Since the Lorentz force is the mechanism through which the electromagnetic field transfers momentum to matter, the absence of sources ($\rho=0, J=0$) immediately causes the force density $f$ to be identically zero throughout $V$. Consequently, the total force $F=\int_V f d \tau$ exerted on the contents of the volume must also be zero, irrespective of whether the electric field $E$ and magnetic field $B$ themselves are zero.
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$\complement\cdots$Counselor
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Definition of Force: The total electromagnetic force $F$ exerted by the electromagnetic field on matter within a volume $V$ is given by the volume integral of the Lorentz force density $f=\rho E+J \times B$.
Source-Free Condition: The problem specifies that the volume $V$ is source-free, meaning there are no free charges ( $\rho=0$ ) or free currents ( $J=0$ ) inside the volume.
Zero Force Density: Under the source-free condition ( $\rho=0, J=0$ ), the Lorentz force density becomes identically zero:
$$ f=(0) E+(0) \times B=0 $$
Conclusion: Since the force density is zero everywhere inside $V$, the total force on the matter (and thus, the force from the field) inside the volume must also be zero:
$$ F=\int_V f d \tau=\int_V 0 d \tau=0 $$
Relevance of Maxwell's Equations: While the simplified Maxwell's equations ( $\nabla \cdot E= 0, \nabla \times E=0, \nabla \cdot B=0, \nabla \times B=0$ ) confirm the existence of a static, source-free field configuration, the result about the total force is a direct consequence of the definition of the Lorentz force and the given condition that $\rho=0$ and $J=0$.
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