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.

🎬Narrated Video

https://youtu.be/Gouo0rclQyQ

🏗️Structural clarification of Poof and Derivation

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