A monochromatic light is incident from air on a refracting surface of a prism of angle 75$$^\circ $$ and refractive index $${n_0} = \sqrt 3 $$. The other refracting surface of the prism is coated by a thin film of material of refractive index n as shown in figure. The light suffers total internal reflection at the coated prism surface for an incidence angle of $$\theta \le 60^\circ $$. The value of n² is .............

A perfectly reflecting mirror of mass M mounted on a spring constitutes a spring-mass system of angular frequency $$\Omega $$ such that $${{4\pi M\Omega } \over h} = {10^{24}}{m^{ - 2}}$$ with h as Planck's constant. N photons of wavelength $$\lambda $$ = 8$$\pi $$ $$ \times $$ 10^{$$ - $$6} m strike the mirror simultaneously at normal incidence such that the mirror gets displaced by 1 $$\mu $$m. If the value of N is x $$ \times $$ 10¹², then the value of x is ................ [Consider the spring as massless]

An optical bench has 1.5 m long scale having four equal divisions in each cm. While measuring the focal length of a convex lens, the lens is kept at 75 cm mark of the scale and the object pin is kept at 45 cm mark. The image of the object pin on the other side of the lens overlaps with image pin that is kept at 135 cm mark. In this experiment, the percentage error in the measurement of the focal length of the lens is ..............

A ball is thrown from ground at an angle $$\theta $$ with horizontal and with an initial speed u₀. For the resulting projectile motion, the magnitude of average velocity of the ball up to the point when it hits the ground for the first time is V₁. After hitting the ground, the ball rebounds at the same angle $$\theta $$ but with a reduced speed of $${{{u_0}} \over \alpha }$$. Its motion continues for a long time as shown in figure. If the magnitude of average velocity of the ball for entire duration of motion is 0.8 V₁, the value of $$\alpha $$ is ..................