Winter 2016 – Q.9

9. If $latex \overline A=\left(y-2xy\right)i+\left(3x+2y\right)j,$ find the circulation of $latex \overline A$ about the circle C in the XY plane with Centre at origin and radius 2, C is traversed in the positive direction. [7M]

We have,

$latex \overline A\left(y-2x\right)i+\left(3x+2y\right)j$

The equation of a circle is given by $latex x^2+y^2=4\;is\;x=2\;\cos\theta,\;y=2\;\sin\theta,\;z=0\;and\;\theta$ varies from 0 to 2π

Now,

$latex \begin{array}{l}\overline r=xi+yj+zk\\\\\;\;\;=2\;\cos\theta\;i+2\;\sin\theta\;j\end{array}$

differentiate w r to θ

$latex \therefore\frac{dr}{d\theta}=-2\;\cos\theta\;i+2\;\sin\theta\;j$

Also

$latex \overline A=\left(2\;\sin\theta-4\;\cos\theta\;\right)i+\left(6\;\cos\theta+4\;\sin\theta\right)j$

$latex \therefore\overline A\;\cdot\frac{dr}{d\theta}=\left[\left(2\;\sin\theta-4\;\cos\theta\;\right)\right]i+\left[\left(6\;\cos\theta+4\;\sin\theta\right)j\right]\cdot\left[-2\;\sin\theta\;i-4\;\cos\theta\;j\right]$

$latex \therefore\overline A\cdot\frac{dr}{d\theta}=\left[-4\;\sin^2\theta-16\;\sin\theta\;\cos\theta+12\;\cos^2\theta\;\right]\;d\theta$

∴ The circulation of $latex \overline A$ about c

$latex \begin{array}{l}=\int_c\;\overline A\cdot dr\\\\=\int_0^{2\pi}\;\overline A\frac{dr}{d\theta}d\theta\\\\=\int_0^{2\pi}\left(-4\;\sin^2\theta-16\;\sin\theta\;\cos\theta+12\;\cos^2\theta\;\right)\;d\theta\\\\=\int_0^{2\pi}\left[-4\left(\frac{1-\cos2\theta}2\right)+8\;\sin2\theta+12\left(\frac{1+\cos\theta}2\right)\right]d\theta\\\\=\int_0^{2\pi}\left(4+8\;\cos2\theta+8\;\sin2\theta\right)d\theta\\\\=\left[4\theta+4\;\sin2\theta-4\;\cos2\theta\right]_0^{2\pi}\\\\=8\pi-4+4\\\\=8\pi\end{array}$

∴ circulation of $latex \overline A$ about c = 8π