📢Zero Total Force for a Source-Free Volume

The total electromagnetic force on the matter inside a volume is determined by the volume integral of the Lorentz force density. The problem specifies a source-free volume, meaning it contains neither free charge nor free current. Since the Lorentz force is the mechanism through which the electromagnetic field transfers momentum to matter, the absence of sources immediately causes the force density to be identically zero throughout this volume. Consequently, the total force exerted on the contents of the volume must also be zero, irrespective of whether the electric field and magnetic field themselves are zero.

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