The electrostatic potential in a charged spherical region of radius \(r\) varies as \(V=a r^3+b\), where \(a\) and b are constants. The total charge in the sphere of unit radius is \(\alpha \times \pi a \in_0\). The value of \(\alpha\) is _________ .

When photon of energy \(4.0\; eV\) strikes the surface of a metal \(\mathrm{A}\), the ejected photoelectrons have maximum kinetic energy \(\mathrm{T}_{\mathrm{A}}\; eV\) end de-Broglie wavelength \(\lambda_{\mathrm{A}} .\) The maximum kinetic energy of photoelectrons liberated from another metal \(\mathrm{B}\) by photon of energy \(4.50 \;eV\) is \(\mathrm{T}_{\mathrm{B}}=\left(\mathrm{T}_{\mathrm{A}}-1.5\right)\;eV\). If the de-Broglie wavelength of these photoelectrons \(\lambda_{B}=2 \lambda_{A}\) then the work function of metal \(\mathrm{B}\) is ............. \(eV\)

A body falling under gravity covers two points \(A\) and \(B\) separated by \(80 \mathrm{~m}\) in \(2 \mathrm{~s}\). The distance of upper point A from the starting point is________ \(\mathrm{m}\) (use \(\mathrm{g}=10 \mathrm{~ms}^{-2}\) )

When the position vector \(\overrightarrow{ r }=x \hat{ i }+y \hat{ j }+z \hat{ k }\) changes sign as \(-\overrightarrow{ r }\), which one of the following vector will not flip under sign change ?

One mole of an ideal monoatomic gas is compressed isothermally in a rigid vessel to double its pressure at room temperature, \(27\,^oC\).The work done on the gas will be

The amplitude of a particle executing \(SHM\) is \(3\,cm\). The displacement at which its kinetic energy will be \(25 \%\) more than the potential energy is: \(.............cm\).

A metal surface is illuminated by a radiation of wavelength \(4500 \; \mathring A\). The ejected photo-electron enters a constant magnetic field of \(2 \; mT\) making an angle of \(90^{\circ}\) with the magnetic field. If it starts revolving in a circular path of radius \(2 \; mm\), the work function of the metal is approximately ............. \(eV\)

First, a set of \({n}\) equal resistors of \(10\; \Omega\) each are connected in series to a battery of emf \(20\; {V}\) and internal resistance \(10\; \Omega .\) A current \(I\) is observed to flow. Then, the \(n\) resistors are connected in parallel to the same battery. It is observed that the current is increased \(20\) times, then the value of \(n\) is .... .

A mercury drop of radius \(10^{-3}\,m\) is broken into \(125\) equal size droplets. Surface tension of mercury is \(0.45\,Nm ^{-1}\). The gain in surface energy is \(......\times 10^{-5}\,J\)

In the given figure an ammeter \(A\) consists of a \(240 \Omega\) coil connected in parallel to a \(10 \Omega\) shunt. The reading of the ammeter is _______ \(\mathrm{mA}\).