- #1
Incand
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Homework Statement
The vector field ##\vec B## is given in spherical coordinates
##\vec B(r,\theta,\phi ) = \frac{B_0a}{r\sin \theta}\left( \sin \theta \hat r + \cos \theta \hat \theta + \hat \phi \right)##.
Determine the line integral integral of ##\vec B## along the curve ##C## with the parametrization ##C: \vec r = (a \cos \alpha, 2a\sin \alpha , \frac{a\alpha}{\pi})## from ##(a,0,0)## to ##(a,0,2a)##.
Homework Equations
Possibly of use
##d\vec r = \sum_1^3 h_i \vec e_i du_i = \frac{d\vec r}{dr}dr +\frac{d\vec r}{d\theta}d\theta +\frac{d\vec r}{d\phi}d\phi##.
3. The Attempt at a Solution
I'm assuming ##C## is given in Cartesian coordinates since nothing else is said. We note that in our parametrization we have ##\alpha \in (0,2\pi)##. We want to calculate
##\int_C \vec F \cdot d\vec r##.
So as I see it I need to either convert the vector field into Cartesian coordinates which looks like a lot of work and probably not the purpose of the exercise or find a way to express the parametrisation in spherical coordinates and then figure out how to integrate that.
One approach would be to get the ##(r,\theta,\phi)## coordinates by the formulas
\begin{cases}
r = \sqrt{x^2+y^2+z^2}\\
\theta = \arccos \frac{z}{\sqrt{x^2+y^2+z^2}}\\
\phi = \arctan \frac{y}{x}
\end{cases}
which seems to give me even worse equations. Any hints on how to get started?