Obtain solution of the free particle problem in two dimensions using Hamilton Jacobi equation and obtain expression for the Hamilton's principal function.
A circular coil is formed from a wire of length $L$ with $n$ turns. The coil carries a current $I$ and is placed in an external uniform magnetic field $B$. Show that maximum torque developed is $\displaystyle\frac{IBL^2}{4n\pi}$.
\(\newcommand{\Prime}{{{\,}^\prime}}\) Approaching the problem of electric field for a uniformly charged sphere in two different ways, find the value of the integral \begin{equation} \vec{\mathcal E} = \frac{1}{4\pi\epsilon_0} \iiint d^3r\Prime \frac{(\vec{r}-\vec {r}\Prime) } {|\vec{r}-\vec{r}\Prime|^3}. \end{equation} where the integral runs over the volume of a sphere of radius \(R\).
