Calculate the value of mean free path \((\lambda)\) for oxygen molecules at temperature \(27^{\circ}\, C\) and pressure \(1.01 \times 10^{5} \,Pa\). Assume the molecular diameter \(0.3 \,nm\) and the gas is ideal. \(\left( k =1.38 \times 10^{-23}\, J\,K ^{-1}\right)\) (in \(nm\))

A parallel plate capacitor whose capacitance \(C\) is \(14\, pF\) is charged by a battery to a potential difference \(V =12\, V\) between its plates. The charging battery is now disconnected and a porcelin plate with \(k =7\) is inserted between the plates, then the plate would oscillate back and forth between the plates with a constant mechanical energy of \(..........pJ\). (Assume no friction)

An electron in a hydrogen atom revolves around its nucleus with a speed of \(6.76 \times 10^6\,ms ^{-1}\) in an orbit of radius \(0.52\,\mathring A\). The magnetic field produced at the nucleus of the hydrogen atom is \(......T\).

Given below are two statements: One is labelled as Assertion \((A)\) and other is labelled as Reason \((R)\). Assertion \((A)\) : Time period of oscillation of a liquid drop depends on surface tension \((S)\), if density of the liquid is \(p\) and radius of the drop is \(r\), then \(T = k \sqrt{ pr ^{3} / s ^{3 / 2}}\) is dimensionally correct, where \(K\) is dimensionless. Reason \((R)\): Using dimensional analysis we get \(R.H.S.\) having different dimension than that of time period. In the light of above statements, choose the correct answer from the options given below.

The temperature of an ideal gas in \(3-\)dimensions is \(300\, {K}\). The corresponding de-Broglie wavelength of the electron approximately at \(300\,{K}\), is \(....\,nm\) \(\left\lfloor{m}_{e}=\text { mass of electron }=9 \times 10^{-31} \,{kg}\right.\) \({h}=\text { Planck constant }=6.6 \times 10^{-34} {Js}\) \(\left.{k}_{{B}}=\text { Boltzmann constant }=1.38 \times 10^{-23}\, {JK}^{-1}\right]\)

Escape velocity of a body from earth is \(11.2 \mathrm{~km} / \mathrm{s}\). If the radius of a planet be one-third the radius of earth and mass be one-sixth that of earth, the escape velocity from the plate is _______.

The coefficient of static friction between a wooden block of mass \(0.5\, kg\) and a vertical rough wall is \(0.2\) The magnitude of horizontal force that should be applied on the block to keep it adhere to the wall will be \(N\) \(\left[ g =10\, ms ^{-2}\right]\)

A plane electromagnetic wave travels in free space along the \(x -\) direction. The electric field component of the wave at a particular point of space and time is \(E =6\; Vm^{-1}\) along \(y -\) direction. Its corresponding magnetic filed component, \(B\) would be