In the given circuit ' \(a\) ' is an arbitrary constant. The value of \(m\) for which the equivalent circuit resistance is minimum, will be \(\sqrt{\frac{ x }{2}}\). The value of \(x\) is ...........

A beam of light has two wavelengths of \(4972\,\mathop A\limits^o \) and \(6216\,\mathop A\limits^o \) with a total intensity of \(3 .6 \times  10^{- 3}\,\,Wm^{-2}\) equally distributed among the two wavelengths. The beam falls normally on an area of \(1\,cm^2\) of a clean metallic surface of work function \( 2.3\,eV.\) Assume that there is no loss of light by reflection and that each capable photon ejects one electron. The number of photoelectrons liberated in \(2\,s\) is approximately

In the given circuit the sliding contact is pulled outwards such that electric current in the circuit changes at the rate of \(8 \mathrm{~A} / \mathrm{s}\). At an instant when R is \(12 \Omega\), the value of the current in the circuit will be ______ A.

When radiation of wavelength \(\lambda \) is incident on a metallic surface, the stopping potential is \(4.8\) volts. If the same surface is illuminated with radiation of double the wavelength, then the stopping potential becomes \(1.6\) volts. Then the threshold wavelength for the surface is

A massless spring gets elongated by amount \(x_1\) under a tension of 5 N . Its elongation is \(x_2\) under the tension of 7 N . For the elongation of \(\left(5 x_1-2 x_2\right)\), the tension in the spring will be,

In the figure shown, after the switch \(‘S’\) is turned from position \(‘A’\) to position \(‘B’\), the energy dissipated in the circuit in terms of capacitance \(‘C’\) and total charge \(‘Q’\) is

Given below are two statements:
Statement I: For a mechanical system of many particles total kinetic energy is the sum of kinetic energies of all the particles.
Statement II: The total kinetic energy can be the sum of kinetic energy of the center of mass w.r.t. to the origin and the kinetic energy of all the particles w.r.t. the center of mass as the reference.
In the light of the above statements,choose the correct answer from the options given below:

A solid sphere of mass \(500\,g\) and radius \(5\,cm\) is rotated about one of its diameter with angular speed of \(10\,rad \, s ^{-1}\). If the moment of inertia of the sphere about its tangent is \(x \times 10^{-2}\) times its angular momentum about the diameter. Then the value of \(x\) will be ..............

Two cylindrical vessels of equal cross sectional area of \(2 \mathrm{~m}^2\) contain water up to height 10 m and 6 m , respectively. If the vessels are connected at their bottom then the work done by the force of gravity is : (Density of water is \(10^3 \mathrm{~kg} / \mathrm{m}^3\) and \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\))

In a Young's double slit experiment, light of \(500\, nm\) is used to produce an interference pattern. When the distance between the slits is \(0.05\, mm\), the angular width (in degree) of the fringes formed on the distance screen is close to\(........^o\)

A thin infinite sheet charge and an infinite line charge of respective charge densities \(+\sigma\) and \(+\lambda\) are placed parallel at \(5\,m\) distance from each other. Points \(P\) and \(Q\), are at \(\frac{3}{\pi} m\) and \(\frac{4}{\pi} m\) perpendicular distance from line charge towards sheet charge, respectively. \(E_P\) and \(E_Q\) are the magnitudes of resultant electric field intensities at point \(P\) and \(Q\), respectively. If \(\frac{E_p}{E_Q}=\frac{4}{a}\) for \(2|\sigma|=|\lambda|\). Then the value of \(a\) is ...........

The mass defect in a particular reaction is \(0.4 \mathrm{~g}\). The amount of energy liberated is \(\mathrm{n} \times 10^7 \mathrm{kWh}\), where\(\mathrm{n}=\)________. (speed of light \(=3 \times 10^8 \mathrm{~m} / \mathrm{s}\) )