In an electron microscope, the resolution that can be achieved is of the order of the wavelength of electrons used. To resolve a width of \(7.5 \times 10^{-12}\, m\), the minimum electron energy required is close to .............. \(keV\)

A hollow spherical shell at outer radius \(R\) floats just submerged under the water surface. The inner radius of the shell is \(r\). If the specific gravity of the shell material is \(\frac{27}{8}\) \(w.r.t.\) water, the value of \(r\) is\(......R\)

Four particles, each of mass \(M\) and equidistant from each other, move along a circle of radius \(R\) under the action of their mutual gravitational attraction. The speed of each particle is

An object viewed from a near point distance of \(25\, {cm}\), using a microscopic lens with magnification \('6',\) gives an unresolved image. A resolved image is observed at infinite distance with a total magnification double the earlier using an eyepiece along with the given lens and a tube of length \(0.6\, {m}\), if the focal length of the eyepiece is euqal to \(.... \,{cm} .\)

A uniformly tapering conical wire is made from a material of Young's modulus \(Y\)  and has a normal, unextended length \(L.\) The radii, at the upper and lower ends of this conical wire, have values \(R\) and \(3R,\)  respectively. The upper end of the wire is fixed to a rigid support and a mass \(M\) is suspended from its lower end. The equilibrium extended length, of this wire, would equal 

In the following circuit, the correct relation between output \(( Y )\) and inputs \(A\) and \(B\) will be

An inductor of inductance \(10\) mH having resistance of \(100\,\Omega\) is connected to battery of E.M.F. \(1.0\) V through a switch as shown in the figure below. After switch is closed, the ratio of instantaneous voltages across the inductor when the current passing through it is \(2\) mA and \(4\) mA is _______.

Two identical springs of spring constant \('2k'\) are attached to a block of mass \(m\) and to fixed support (see figure). When the mass is displaced from equilibrium position on either side, it executes simple harmonic motion. The time period of oscillations of this sytem is ...... .

A Copper \((Cu)\) rod of length \(25\, {cm}\) and cross- sectional area \(3\, {mm}^{2}\) is joined with a similar Aluminium \((Al)\) rod as shown in figure. Find the resistance of the combination between the ends \(A\) and \(B\) (in \({m} \Omega\)) (Take Resistivity of Copper \(=1.7 \times 10^{-8}\, \Omega \,{m}\), Resistivity of Aluminium \(=2.6 \times 10^{-8}\, \Omega \,{m}\) )

From the following combinations of physical constants (expressed through their usual symbols) the only combination, that would have the same value in different systems of units, is

A block of mass 2 kg is attached to one end of a massless spring whose other end is fixed at a wall. The spring-mass system moves on a frictionless horizontal table. The spring's natural length is 2 m and spring constant is \(200 \mathrm{~N} / \mathrm{m}\). The block is pushed such that the length of the spring becomes 1 m and then released. At distance \(\mathrm{x} \mathrm{m}(\mathrm{x} \lt 2)\) from the wall. the speed of the block will be :