For a charged spherical ball, electrostatic potential inside the ball varies with \(r\) as \(V =2 ar ^2+ b\). Here, \(a\) and \(b\) are constant and \(r\) is the distance from the center. The volume charge density inside the ball is \(-\lambda a \varepsilon\). The value of \(\lambda\) is \(...........\). \(\varepsilon=\) permittivity of medium.

Two concentric circular coils with radii \(1\,cm\) and \(1000\,cm\), and number of turns \(10\) and \(200\) respectively are placed coaxially with centers coinciding. The mutual inductance of this arrangement will be \(.........\times 10^{-8}\,H\)(Take, \(\pi^2=10\))

A small spherical ball of radius \(0.1 \,mm\) and density \(10^{4} \,kg m ^{-3}\) falls freely under gravity through a a distance \(h\) before entering a tank of water. If after entering the water the velocity of ball does not change and it continue to fall with same constant velocity inside water, then the value of \(h\) wil be \(m\). (Given \(g =10 \,ms ^{-2}\), viscosity of water \(=1.0 \times 10^{-5} \,N - sm ^{-2}\) )

When vector \(\overrightarrow{ A }=2 \hat{ i }+3 \hat{ j }+2 \hat{ k }\) is subtracted from vector \(\vec{B}\), it gives a vector equal to \(2 \hat{j}\). Then the magnitude of vector \(\vec{B}\) will be:

For a given gas at \(1\,atm\) pressure, \(rms\) speed of the molecules is \(200\,m/s\) at \(127\,^oC.\) At \(2\,atm\) pressure and at \(227\,^oC,\) the \(rms\) speed of the molecules will be

A charge \(Q\) is distributed over three concentric spherical shell of radii \(a, b, c (a < b < c)\) such that their surface charge densities are equal to one another. The total potential at a point at distance \(r\) from their common centre, where \(r < a\), would be

If \('f^{\prime}\) denotes the ratio of the number of nuclei decayed \(\left(N_{d}\right)\) to the number of nuclei at \(t=0\) \(\left({N}_{0}\right)\) then for a collection of radioactive nuclei, the rate of change of \('f'\) with respect to time is given as: \([\lambda\) is the radioactive decay constant]

Choose the correct option:

\(\sigma\) is the uniform surface charge density of a thin spherical shell of radius \(R\). The electric field at any point on the surface of the spherical shell is _______.