A ray of light \(AO\) in vacuum is incident on a glass slab at angle \(60^o\) and refracted at angle \(30^o\) along \(OB\) as shown in the figure. The optical path length of light ray from \(A\) to \(B\) is

Suppose two planets (spherical in shape) of radii \({R}\) and \(2 {R}\), but mass \({M}\) and \(9\, {M}\) respectively have a centre to centre separation \(8\, {R}\) as shown in the figure. A satellite of mass \('{m}'\) is projected from the surface of the planet of mass \('M'\) directly towards the centre of the second planet. The minimum speed \('v'\) required for the satellite to reach the surface of the second planets is \(\sqrt{\frac{a}{7} \frac{G M}{R}}\) then the value of \('a'\) is \(....\) [Given: The two planets are fixed in their position]

Two particles move at right angle to each other. Their de Broglie wavelengths are \({\lambda _1}\) and \({\lambda _2}\) respectively. The particles suffere perfectly inelastic collision. The de Broglie wavelength \(\lambda \) , of the final particle, is given by

In a Young's double slit experiment, an angular width of the fringe is \(0.35^{\circ}\) on a screen placed at \(2\,m\) away for particular wavelength of \(450\,nm\). The angular width of the fringe, when whole system is immersed in a medium of refractive index \(7 / 5\), is \(\frac{1}{\alpha}\). The value of \(\alpha\) is ..............

A piston of mass \(M\) is hung from a massless spring whose restoring force law goes as \(\mathrm{F}=-\mathrm{kx}^3\), where k is the spring constant of appropriate dimension. The piston separates the vertical chamber into two parts, where the bottom part is filled with ' \(n\) ' moles of an ideal gas. An external work is done on the gas isothermally (at a constant temperature T) with the help of a heating filament (with negligible volume) mounted in lower part of the chamber, so that the piston goes up from a height \(\mathrm{L}_0\) to \(\mathrm{L}_1\), the total energy delivered by the filament is (Assume spring to be in its natural length before heating)

An insulated wire is wound so that it forms a flat coil with \(N = 200\) turns. The radius of the innermost turn is \(r_1 = 3\) cm, and of the outermost turn \(r_2 = 6\) cm. If \(20\) mA current flows in it then the magnetic moment will be \(\alpha \times 10^{-2}\) A.m\(^2\). The value of \(\alpha\) is _____.

Two vertical parallel mirrors \(A\) and \(B\) are separated by \(10\,cm\). A point object \(O\) is placed at a distance of \(2\,cm\) from mirror \(A\). The distance of the second nearest image behind mirror A from the mirror \(A\) is \(.........\,cm\)

In interference experiment the path difference between two interfering waves at a point \(A\) on the screen is \(\lambda/3\), where \(\lambda\) is the wavelength of these waves, and at another point \(B\) the path difference is \(\lambda/6\). The ratio of intensities at points \(A\) and \(B\) is _______.

A ray of light entering from air into a denser medium of refractive index \(\frac{4}{3}\), as shown in figure. The light ray suffers total internal reflection at the adjacent surface as shown. The maximum, value of angle \(\theta\) should be equal to -

The temperature of a gas is \(-78^{\circ} \mathrm{C}\) and the average translational kinetic energy of its molecules is \(\mathrm{K}\). The temperature at which the average translational kinetic energy of the molecules of the same gas becomes \(2 \mathrm{~K}\) is _______.

A parallel plate capacitor with plate area \(A\) and plate separation \(d\) is filled with a dielectric material of dielectric constant \(K =4\). The thickness of the dielectric material is \(x\), where \(x < d\). Let \(C_1\) and \(C_2\) be the capacitance of the system for \(x =\frac{1}{3} d\) and \(x =\frac{2 d }{3}\), respectively. If \(C _1=2 \mu F\) the value of \(C _2\) is \(........... \mu F\)

Proton with kinetic energy of \(1\;MeV\) moves from south to north. It gets an acceleration of \(10^{12}\; \mathrm{m} / \mathrm{s}^{2}\) by an applied magnetic field (west to east). The value of magnetic field :.......\(mT\) (Rest mass of proton is \(1.6 \times 10^{-27} \;\mathrm{kg}\) )

Five positive charges each having charge q are placed at the vertices of a pentagon as shown in the figure. The electric potential (V) and the electric field \((\vec{E})\) at the center O of the pentagon due to these five positive charges are: