I am working on a problem that asks to find the magnetic field at the origin of a logarithmic spiral $r = e^\theta$ from $\theta = 0$ to $\theta = 2\pi$, where the angle $\theta$ is measured counterclockwise from the positive x axis.
I was able to do some research and found the following information about the curve.
I know that the tan() of the angle between a radial line and the tangent is 1 in this case because the coefficient on the exponent is 1.
My question is how the differential length $ds$ was determined? Also, I have no idea how $tan \beta = r d\theta / dr$. Can anyone help explain the geometry on this problem to me?
Thanks so much for your help.
Best Answer
$dr$ is the infinitesimal change in the radial direction while $r d\theta$ is the corresponding change in the perpendicular direction. The arc length is therefore: $$ ds = \sqrt{dr^2 + (rd\theta)^2} = \\ \sqrt{1 + \left(r\frac{d\theta}{dr}\right)^2}dr $$ Also, $tan\beta$ is the ratio of tangential to the radial displacement i.e. $$ tan\beta = \frac{rd\theta}{dr} $$