A light ray is incident on a glass slab of thickness $$4 \sqrt{3} \mathrm{~cm}$$ and refractive index $$\sqrt{2}$$ The angle of incidence is equal to the critical angle for the glass slab with air. The lateral displacement of ray after passing through glass slab is ______ $$\mathrm{cm}$$.

(Given $$\sin 15^{\circ}=0.25$$)

Two wires $$A$$ and $$B$$ are made up of the same material and have the same mass. Wire $$A$$ has radius of $$2.0 \mathrm{~mm}$$ and wire $$B$$ has radius of $$4.0 \mathrm{~mm}$$. The resistance of wire $$B$$ is $$2 \Omega$$. The resistance of wire $$A$$ is ________ $$\Omega$$.

The displacement of a particle executing SHM is given by $$x=10 \sin \left(w t+\frac{\pi}{3}\right) m$$. The time period of motion is $$3.14 \mathrm{~s}$$. The velocity of the particle at $$t=0$$ is _______ $$\mathrm{m} / \mathrm{s}$$.

A parallel plate capacitor of capacitance $$12.5 \mathrm{~pF}$$ is charged by a battery connected between its plates to potential difference of $$12.0 \mathrm{~V}$$. The battery is now disconnected and a dielectric slab $$(\epsilon_{\mathrm{r}}=6)$$ is inserted between the plates. The change in its potential energy after inserting the dielectric slab is ________ $$\times10^{-12} \mathrm{~J}$$.