# Direction of Integration in Biot Savart's Law (Line Integral)

Let's say we have a loop with a clockwise current, and my angle increases in the counter-clockwise direction (that is, $$\hat{\phi}$$ is counter-clockwise).

I have $$B=\frac{\mu_{0} i\vec{dl} \times \vec{r}}{4 \pi |\vec{r}|^{3}}$$

If I were to evaluate the line integral counter-clockwise, then when I change this integral in terms of $$\phi$$, my integral goes from $$0$$ to $$2 \pi$$.

If I evaluatethis integral clockwise, when I switch this integral in terms of $$\phi$$, my integral goes from $$2 \pi$$ to $$0$$ $$-$$ but now my $$\vec{dl}$$ is $$-R d\phi \hat{\phi}$$, so the two negative signs just compensate.

However, in one case, the current is in the same direction as $$\vec{dl}$$ and in the other, it is antiparallel, so it looks like I will get two different answers (although i expect the same answer)!

Which, of course, just leaves me confused.

Thanks for any help!

• Why do you think the direction of the magnetic field should depend on the mechanics of the integration? The current itself does not change direction. – Triatticus May 5 '19 at 16:14
• My question wasn't very clear, sorry! I've edited it a bit. – Rahul Arvind May 5 '19 at 16:27

The actual Biot-Savart's law for a wire carrying a steady current reads

$$B(\vec{r})=\frac{\mu_0}{4\pi}\int\frac{\vec{I}\times\vec{r'}}{r'^3}d\ell'$$

where $$\vec{r'}$$ is a vector pointing from the charge to the point in space $$\vec{r}$$. What you have written is a simplification that we can always make for the steady current carrying wire because $$I$$ is just some constant. So we define a vector $$d\ell'$$ that satisfies the condition

$$\vec{d\ell'}=d\ell \hat{I}$$

where

$$\hat{I} = \frac{\vec{I}}{|{I}|}$$

and thus it is equivalent to write this as

$$B(\vec{r})=\frac{\mu_0 I}{4\pi}\int\frac{\vec{d\ell'}\times\vec{r'}}{r'^3}$$

which is convenient. The answer for your question is then that it absolutely matters which way you choose to integrate: you must integrate in the direction that the current flows. You're correct to think that the direction is not arbitrary.

• Thanks for the detailed response! – Rahul Arvind May 8 '19 at 13:01

The convention is to parametrize $$\vec{d\ell}$$ in the direction that the current is flowing so that there is no ambiguity. When you stick to this convention the sign ambiguity goes away.