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A point charge \(q\) is embedded at the center of a sphere with dielectric constant \(\kappa_1\). The sphere is itself embedded in an infinite volume with dielectric constant \(κ_2\). Find the electric field out side the sphere and surface charge density on  the surface of the sphere.

Question : For a particle moving in a potential well $$\displaystyle V(x) = -V_0\,\text{sech}^2(\alpha x),\qquad -V_0 < E < 0$$  find the time period as a function of energy of the particle. 

Answer : $ \displaystyle ({\pi\over \alpha})\sqrt{(2m/|E|)}$

Question : A particle in one dimension is subject to a force
$$ F = -k x +{ \lambda\over x^3} $$
Find the potential energy and the exact period of oscillations for
$\lambda = 6ka^4, E= 5k a^2$.

Question: Show that the time period of a relativistic oscillator as  function of amplitude $a$ is given by $$ T\approx \frac{2\pi}{\omega}\left[ 1 + \frac{3\omega^2a^2}{16 c^2} + \cdots \right] $$ where the restoring force is $- kx$ and $k=m \omega2 $.