A symmetric star shaped conducting wire loop is carrying a steady state current $${\rm I}$$ as shown in the figure. The distance between the diametrically opposite vertices of the star is $$4a.$$ The magnitude of the magnetic field at the center of the loop is

A photoelectric material having work-function $${\phi _0}$$ is illuminated with light of wavelength $$\lambda \left( {\lambda < {{he} \over {{\phi _0}}}} \right).$$ The fastest photoelectron has a de-Broglic wavelength $${\lambda _d}.$$ A change in wavelength of the incident light by $$\Delta \lambda $$ result in a change $$\Delta {\lambda _d}$$ in $${\lambda _d}.$$ Then the ratio $$\Delta {\lambda _d}/\Delta \lambda $$ is proportional to

Consider regular polygons with number of sides $$n=3,4,5....$$ as shown in the figure. The center of mass of all the polygons is at height $$h$$ from the ground. They roll on a horizontal surface about the leading vertex without slipping and sliding as depicted. The maximum increase in height of the locus of the center of mass for each polygon is $$\Delta $$. Then $$\Delta $$ depends on $$n$$ and $$h$$ as

Consider an expanding sphere of instantaneous radius R whose total mass remains constant. The expansion is such that the instantaneous density $$\rho $$ remains uniform throughout the volume. The rate of fractional change in density $$\left( {{1 \over \rho } {{d\rho } \over {dt}}} \right)$$ is constant. The velocity $$v$$ of any point on the surface of the expanding sphere is proportional to