A charge $$+Q$$ is uniformly distributed throughout the volume of a dielectric sphere of radius $$R$$ and dielectric constant $$'{\varepsilon _R}'.$$ Based on Gauss law, determine the expressions for the electric field $$E$$ as a function of distance $$'r'$$ from the center of the sphere, within the ranges $$0 < r < R$$ and $$R \le r.$$ Indicate expression $$(s)$$ for the critical point $$(s)$$ on the sketch.

Two parallel wires separated by a distance $$'d'$$ are carrying a current $$'{\rm I}'$$ in the same direction. The magnetic field along a line running parallel to these wires at midway between them

If $$A = \left[ {\matrix{ 1 & { - 2} & { - 1} \cr 2 & 3 & 1 \cr 0 & 5 & { - 2} \cr } } \right]$$ and $$adj (A)$$ $$ = \left[ {\matrix{ { - 11} & { - 9} & 1 \cr 4 & { - 2} & { - 3} \cr {10} & k & 7 \cr } } \right]$$ then $$k=$$

Find the eigen values and eigen vectors of the matrix $$\left[ {\matrix{ 3 & { - 1} \cr { - 1} & 3 \cr } } \right]$$