# 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.

{% embed url="<https://youtu.be/Gouo0rclQyQ>" %}

{% embed url="<https://viadean.notion.site/Total-Electromagnetic-Force-in-a-Source-Free-Static-Volume-28a1ae7b9a3280718bd2ed75e8b2df79?source=copy_link>" %}