If the maximum value of accelerating potential provided by a ratio frequency oscillator is \(12\, {kV}\). The number of revolution made by a proton in a cyclotron to achieve one sixth of the speed of light is ....... . \(\left[{m}_{{p}}=1.67 \times 10^{-27} {kg}, {e}=1.6 \times 10^{-19} {C},\right.\) Speed of light \(\left.=3 \times 10^{8} {m} / {s}\right]\)

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)

A small bob tied at one end of a thin string of length \(1\, m\) is describing a vertical circle so that the maximum and minimum tension in the string are in the ratio \(5: 1 .\) The velocity of the bob at the height position is ............. \(m / s\) (Take \(\left.g=10 \,m / s ^{2}\right)\)

The ratio of average electric energy density and total average energy density of electromagnetic wave is:

Three straight parallel current carrying conductors are shown in the figure. The force experienced by the middle conductor of length \(25\,cm\) is

A heat source at \(T = 10^3\, K\) is connected to another heat reservoir at \(T = 10^2\, K\) by a copper slab which is \(1\, m\) thick. Given that the thermal conductivity of copper is \(0.1\, WK^{-1}\, m^{-1}\), the energy flux through it in the steady state is ........... \(Wm^{-2}\)

The root mean square velocity of molecules of gas is

A vertical electric field of magnitude \(4.9 \times 10^{5} N / C\) just prevents a water droplet of a mass \(0.1\, g\) from falling. The value of charge on the droplet will be  ........ \(\times 10^{-9} \;C\) \(\left(\right.\) Given \(\left.g =9.8 m / s ^{2}\right)\)

What is the lateral shift of a ray refracted through a parallel-sided glass slab of thickness ' \(h\) ' in terms of the angle of incidence ' \(i\) ' and angle of refraction ' \(r\) ', if the glass slab is placed in air medium?

A prism of angle \(A=1^{\circ}\) has a refractive index \(\mu=1.5 .\) A good estimate for the minimum angle of deviation (in degrees) is close to \(N/10\) Value of \(N\) is......

Given below are two statements: Statement\(- I\) : Susceptibilities of paramagnetic and ferromagnetic substances increase with decrease in temperature. Statement\(- II\): Diamagnetism is a result of orbital motions of electrons developing magnetic moments opposite to the applied magnetic field. Choose the \(CORRECT\) answer from the options given below : -

If the charge on a capacitor is increased by \(2C\), the energy stored in it increases by \(44 \%\). The original charge on the capacitor is (in \(C\) )

Two identical symmetric double convex lenses of focal length \(f\) are cut into two equal parts \(L_1, L_2\) by \(A B\) plane and \(L_3, L_4\) by \(X Y\) plane as shown in figure respectively. The ratio of focal lengths of lenses \(L_1\) and \(L_3\) is ________.