Question A.4.3: Evaluate the integral ∮s A· ds where A(x, y) = αrr + βzz, an...
Evaluate the integral \oint_{s} A· ds where A(x, y) = αr\hat{r} + βz\hat{z}, and S is the closed surface of the cylinder shown in Fig. A.8. Here, α and β are constants.
We use cylindrical coordinates. First, we determine the unit normals on the surface of the cylinder. We find that
\hat{n} = \left\{\begin{matrix} \hat{z} \qquad z = L\\ \hat{r} \qquad r = a\\ -\hat{z} \qquad z = 0. \end{matrix} \right.
Next, we evaluate A· ds over S using Eq. (A.17) and obtain
Eq. (A.17): ds = \left\{\begin{matrix} r dΦ dz \qquad (constant r) \\ dr dz \qquad (constant Φ) \\ r dr dΦ \qquad (constant z),\end{matrix} \right.
A· ds = \left\{\begin{matrix} βLr dr dΦ \qquad (ds = r dr dΦ)\qquad z = L\\ α a^{2}dΦ dz \qquad (ds = a dΦ dz) \qquad r = a\\ 0 \qquad (ds = r dr dΦ)\qquad z = 0. \end{matrix} \right.
Therefore,
\int_{s} A· ds = βL \int_{0}^{2π} \int_{0}^{a} r dr dΦ + α a^{2}\int_{0}^{L} \int_{0}^{2π} dΦ dz
= βL πa^{2} + αa^{2}2πL
= (2α+β)πa^{2}L.