A wire carrying a current $I$ is bent into the shape of an exponential spiral, $r = e^θ$, from $\theta = 0$ to $\theta = 2\pi$ as shown in the figure below.
To complete a loop, the ends of the spiral are connected by a straight wire at the origin along the x axis. Find the magnitude $G$ and direction of the magnetic field $B$.
Hint: Use the Biot–Savart law. The angle $\beta$ between a radial line and its tangent line at any point on the curve $r$ = $f (θ)$ is related to the function in the following way: $tan β = \frac{r}{\frac{dr}{d\theta}}$.
The answer and how to do it can be found here at page 7: Full question
The thing is that I solved it a different way and came to the same solution. I first changed the coordinate system into polar coordinates. So the vector along the spiral can be written as: $L = (r\cos(θ),r\sin(θ),0) = (e^θ\cos(θ),e^θ\sin(θ)).$
After calculating the cross product and the norm of $-L$, using Biot-Savart I was able to come to the same solution as the one shown in the question. But this only holds if I don't use the Jacobian determinant for polar coordinates which is $r = e^θ$. So,my question is am I doing it wrong, do we need the Jacobian determinant when changing to polar coordinates for the law of Biot-Savart?
Thanks in advance.