Number of molecules in a volume of \(4\, cm ^{3}\) of a perfect monoatomic gas at some temperature \(T\) and at a pressure of \(2\, cm\) of mercury is close to \(?\) (Given, mean kinetic energy of a molecule (at \(T\) ) is \(4 \times 10^{-14}\) erg, \(g=980\, cm / s ^{2}\), density of mercury \(=13.6\, g / cm ^{3}\))

The quantities \(\quad x=\frac{1}{\sqrt{\mu_{0} \epsilon_{0}}}, y=\frac{E}{B}\) and \(z=\frac{l}{C R}\) are defined where \(C-\) capacitance \(R-\)Resistance, \(l-\)length, \(E-\)Electric field, \(B-\)magnetic field and \(\varepsilon_{0}, \mu_{0},\) -free space permittivity and permeability respectively. Then....

Two identical metallic spheres \(A\) and \(B\) when placed at certain distance in air repel each other with a force of \(F\). Another identical uncharged sphere \(C\) is first placed in contact with \(A\) and then in contact with \(B\) and finally placed at midpoint between spheres \(A\) and \(B\). The force experienced by sphere \(C\) will be.

Read the following statements: \(A.\) When small temperature difference between a liquid and its surrounding is doubled the rate of loss of heat of the liquid becomes twice. \(B.\) Two bodies \(P\) and \(Q\) having equal surface areas are maintained at temperature \(10^{\circ}\,C\) and \(20^{\circ}\,C\). The thermal radiation emitted in a given time by \(P\) and \(Q\) are in the ratio \(1: 1.15\) \(C.\) A carnot Engine working between \(100\,K\) and \(400\,K\) has an efficiency of \(75 \%\) \(D.\) When small temperature difference between a liquid and its surrounding is quadrupled, the rate of loss of heat of the liquid becomes twice. Choose the correct answer from the options given below :

A bakelite beaker has volume capacity of \(500\, cc\) at \(30^{\circ} C\). When it is partially filled with \(V _{ m }\) volume (at \(30^{\circ}\) ) of mercury, it is found that the unfilled volume of the beaker remains constant as temperature is varied. If \(\gamma_{\text {(beaker) }}=6 \times 10^{-6}{ }^{\circ} C ^{-1}\) and \(\gamma_{(\text {mercury })}=1.5 \times 10^{-4}{ }^{\circ} C ^{-1},\) where \(\gamma\) is the coefficient of volume expansion, then \(V _{ m }(\)in \(cc )\) is close to

A metallic rod oflength '\(l\)' is tied to a string oflength \(2l\). and made to rotate with angular speed w on a horizontal table with one end of the string fixed. Ifthere is a vertical magnetic field '\(B\)' in the region, the \(e.m.f.\) induced across the ends of the rod is

A bob of mass \(m\) attached to an inextensible string of length \(l\) is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed \(\omega\ rad/s\) about the vertical. About the point of suspension:

In a long glass tube, mixture of two liquids A and B with refractive indices 1.3 and 1.4 respectively, forms a convex refractive meniscus towards \(A\). If an object placed at 13 cm from the vertex of the meniscus in A forms an image with a magnification of ' \(-2^{\prime}\) then the radius of curvature of meniscus is :

A liquid of density \(600\) kg/m\(^3\) flowing steadily in a tube of varying cross-section. The cross-section at a point \(A\) is \(1.0\) cm\(^2\) and that at \(B\) is \(20\) mm\(^2\). Both the points \(A\) and \(B\) are in same horizontal plane, the speed of the liquid at \(A\) is \(10\) cm/s. The difference in pressures at \(A\) and \(B\) points is ________ Pa.

A girl standing on road holds her umbrella at \(45^{\circ}\) with the vertical to keep the rain away. If she starts running without umbrella with a speed of \(15 \sqrt{2} \; kmh ^{-1}\), the rain drops hit her head vertically. The speed of rain drops with respect to the moving girl is ...........  \(kmh ^{-1}\)

The energy associated with electric field is \((U_E)\) and with magnetic field is \((U_B)\) for an electromagnetic wave in free space. Then