🧄Total Electromagnetic Force in a Source-Free Static Volume

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.

Total Electromagnetic Force in a Source-Free Static Volume | Proof and Derivationviadean on Notion