Induced Voltage Exercise (Part 2)

The area element in polar co-ordinates is $rdrd\theta$ . Now, for the magnetic flux, we have

$\phi\ =\ \int\left(t^{2}x^{2}+t^{3}\cos y\right)\cdot dA$

$=\int_{0}^{2\pi}\int_{0}^{1}\left(t^{2}\left(r\cos\theta\right)^{2}+t^{3}\cos\left(r\sin\theta\right)\right)rdrd\theta$

Now, if we denote $a\ =\ \int_{0}^{2\pi}\int_{0}^{1}r^{3}\cos^{2}\theta drd\theta$ and $b=\int_{0}^{2\pi}\int_{0}^{1}r\cos\left(r\sin\theta\right)drd\theta$ , then the induced voltage (using Lenz's Law ) comes out to be

$V\left(t\right)=-\left(2at\ +\ 3t^{2}b\right)$

So, we have $\left|V\left(2\right)\right|=36.3206$

Note that $a$ can be evaluated manually but $b$ needs a numerical approximation.

Nice Problem. Please increase the difficulty of the question.