Unit vector in $ds$ substitution (while changing coordinate systems)

Question

I had to calculate the line integral of a homogenous vector field $E=E.e_y$ from an angle $\phi=o$ to $\phi=\phi_{e}$ having a radius $\rho$. the formula normally is $\varphi=\int E \space ds$ but since the coordinate system used is cylindrical I converted $ds$ to $R\ dR\ d\phi$. However, the answer I got was wrong and in the provided solution. $ds$ was substituted by $Rd\varpThe session negotiation failed.

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Answer 1

The issue is with the notation $d\mathbf{s}$ is not a surface element, so it's not $Rdrd\phi$. It is a length, tangent to the curve. You need to decompose it into $ds_x$ and $ds_y$. The one that you care about is $ds_y$, since $ds_x\mathbf{e}_x\cdot\mathbf{E}=0$.

You can write $ds_y$ in terms of the angle: $$ds_y=\rho\cos\phi d\phi$$ See if this solves the problem.

Answer 2

Your curve is parametrized as

$$\mathbf{s}(\phi) = \rho(\cos\phi,\sin\phi,0),$$

hence

$$\mathrm{d}\mathbf{s} = \rho(-\sin\phi,\cos\phi,0)\,\mathrm{d}\phi.$$

Combined with $\mathbf{E} = (0,E,0)$, we get

$$\int\mathbf{E}\cdot\mathrm{d}\mathbf{s} = \int_0^{\phi_e}E\rho\cos\phi\,\mathrm{d}\phi = E\rho \sin\phi_e.$$