The volume charge density of a sphere of radius \(6 \,m\) is \(2 \,\mu cm ^{-3}\). The number of lines of force per unit surface area coming out from the surface of the sphere is \(....\times 10^{10}\, NC ^{-1}\).  [Given : Permittivity of vacuum  \(\left.\epsilon_{0}=8.85 \times 10^{-12} C ^{2} N ^{-1}- m ^{-2}\right]\)

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 

A \(2\, \mu F\) capacitor \(C _{1}\) is first charged to a potential difference of \(10\, V\) using a battery.Then the battery is removed and the capacitor is connected to an uncharged capacitor \(C _{2}\) of \(8\, \mu F\). The charge in \(C _{2}\) on equilibrium condition is \(\ldots\,\mu C\). (Round off to the Nearest Integer)

The ratio of speeds of electromagnetic waves in vacuum and a medium, having dielectric constant k=3 and permeability \(\mu=2\mu_{0}\), is
( \(\mu_0=\) permeability of vacuum)

The \(Q\)-value of a nuclear reaction and kinetic energy of the projectile particle, \(K _{ p }\) are related as

Two masses \(m\) and \(\frac{m}{2}\) are connected at the two ends of a massless rigid rod of length \(l\). The rod is suspended by a thin wire of torsional constant \(k\) at the centre of mass of the rodmass system (see figure). Because of torsional constant \(k\), the restoring torque is \(\tau  = k\,\theta \) for angular displacement \(\theta \). If the rod is rotated by \(\theta _0\) and released, the tension in it when it passes through its mean position will be

Nearly \(10 \%\) of the power of a \(110\,W\) light bulb is converted to visible radiation. The change in average intensities of visible radiation, at a distance of \(1\, m\) from the bulb to a distance of \(5\,m\) is \(a \times 10^{-2}\,W / m ^{2}\). The value of ' \(a\) ' will be.

When a particle executes \(SHM\) the nature of graphical representation of velocity as a function of displacement is :

Sound travels in a mixture of two moles of helium and n moles of hydrogen. If rms speed of gas molecules in the mixture is \(\sqrt{2}\) times the speed of sound, then the value of \(n\) will be

Given below are two statements : Statement \((I)\) : The mean free path of gas molecules is inversely proportional to square of molecular diameter. Statement \((II)\) : Average kinetic energy of gas molecules is directly proportional to absolute temperature of gas. In the light of the above statements, choose the correct answer from the option given below:

A series \(LCR\) circuit is connected to an ac source of \(220\,V , 50 Hz\). The circuit contain a resistance \(R=100 \Omega\) and an inductor of inductive reactance \(X_L=79.6 \Omega\). The capacitance of the capacitor needed to maximize the average rate at which energy is supplied will be \(..........\mu F\).

A photon of wavelength \(\lambda \) is scattered from an electron, which was at rest. The wavelength shift \(\Delta \lambda \) is three times of \( \lambda \) and the angle of scattering \(\theta \) is \(60^o\) . The angle at which the electron recoiled is \(\phi \). The value of \(tan\,\phi \) is : ( electron speed is much smaller than the speed of light)

The density of a solid metal sphere is determined by measuring its mass and its diameter. The maximum error in the density of the sphere is \(\left(\frac{x}{100}\right) \% .\) If the relative errors in measuring the mass and the diameter are \(6.0 \%\) and \(1.5 \%\) respectively, the value of \(x\) is