[Physics] Volume integral of current density

electromagnetismintegrationmagnetic fieldsmagnetostaticsVector Fields

I'm currently studying magnetostatics and have a simple question :

What is the volume integral of the current density over the whole space in magnetostatics $$\int_{V} \textbf{j} \space d^3\textbf{r}$$ in general? Can I just break up the integration into $$\int \textbf{j}\space {dS}\space {dr}=\int0 \space dr =0,$$ since: $$\int_S \textbf{j} =0,$$ if $S$ is a closed surface?

Best Answer

If your main motive is to show that, in magnetostatics, $$\int\limits_{V}\mathbf{J}\, d\tau =0 $$ Then there is actually quite a simple way to achieve that, using the fact that, in magnetostatics, $$\frac{\partial\rho}{\partial t}=0 $$ We start with the relation, $$\int\limits_{V}\mathbf{J}\, d\tau =\frac{d\mathbf{p}}{dt} $$ Where, $\mathbf{p} $ is the electric dipole. Also we know that, $$\mathbf{p}=\int\limits_{V}\rho\mathbf{r}\, d\tau $$ Therefore, $$\frac{d\mathbf{p}}{dt}=\int\limits_{V}\frac{\partial\rho}{\partial t}\mathbf{r}\, d\tau=0 $$ Which makes, $$\int\limits_{V}\mathbf{J}\, d\tau =0 $$