How would one go about computing the vector line integral presented in the Biot-Savart law: $$\vec{B}=\int_c\frac{\mu_0I}{4\pi} \frac{d\vec{l}\times\hat{r}}{r^2}$$ I know how to compute vector line integrals: $\int_c\vec{F}\cdot d\vec{s} = \int_a^b\vec{F}(\vec{r}(t))\cdot (\vec{r}(t))'dt$, but I have no idea where to start with the above integral. Any help would be greatly appreciated.
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$\begingroup$ @TedShifrin that's what I'm struggling to figure out... $\endgroup$– JBatswaniCommented Oct 8, 2023 at 0:57
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$\begingroup$ The difference here is the answer is a vector rather than a scalar. What line integral is the $\hat i$-component of that vector? $\endgroup$– Ted ShifrinCommented Oct 8, 2023 at 1:08
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$\begingroup$ @TedShifrin so how would I go about solving this? $\endgroup$– JBatswaniCommented Oct 8, 2023 at 4:58
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$\begingroup$ Exactly the same way : choose a parametrization $\vec{l}(t)$ for the line $c$, hence $\mathrm{d}\vec{l} = \vec{l'}\mathrm{d}t$. $\endgroup$– AbezhikoCommented Oct 8, 2023 at 7:14
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