The energy stored in the electric field produced by a metal sphere is \(4.5\, J\). lf the sphere contains \(4\,\mu C\) charge, its radius will be.......\(mm\) : [Take : \(\frac{1}{{4\pi {\varepsilon _0}}} = 9 \times {10^9}\,N - {m^2}\,/{C^2}\, ]\)

If \(\vec{a}\) and \(\vec{b}\) makes an angle \(\cos ^{-1}\left(\frac{5}{9}\right)\) with each other, then \(|\vec{a}+\vec{b}|=\sqrt{2}|\vec{a}-\vec{b}|\) for \(|\vec{a}|=n|\vec{b}|\) The integer value of \(n\) is _______.

A small square loop of side \('a'\) and one turn is placed inside a larger square loop of side \({b}\) and one turn \((b \gg a)\). The two loops are coplanar with thei centres coinciding. If a current \(I\) is passed in the square loop of side \('b',\) then the coefficient of mutual inductance between the two loops is

Two small equal point charges of magnitude \(q\) are suspended from a common point on the ceiling by insulating mass less strings of equal lengths. They come to equilibrium with each string making angle \(\theta \) from the vertical. If the mass of each charge is \(m,\) then the electrostatic potential at the centre of line joining them will be \(\left( {\frac{1}{{4\pi { \in _0}}} = k} \right).\)

Ratio of thermal energy released in two resistor \(R\) and \(3 R\) connected in parallel in an electric circuit is :

A uniform cylindrical rod of length \(L\) and radius \(r\), is made from a material whose Young's modulus of Elasticity equals \(Y\). When this rod is heated by temperature \(T\) and simultaneously subjected to a net longitudinal compressional force \(F\), its length remains unchanged. The coefficient of volume expansion, of the material of the rod, is (nearly) equals to

An electromagnetic wave of frequency 100 MHz propagates through a medium of conductivity, \(\sigma=10 mho / m\). The ratio of maximum conducting current density to maximum displacement current density is _________ .
\(\left[\right.\) Take \(\left.\frac{1}{4 \pi \epsilon_0}=9 \times 10^9 Nm ^2 / C ^2\right]\)

Match List \(− I\) (Event) with List \(−II \) (Order of the time interval for happening of the event) and select the correct option from the options given below the lists

List \(- I\)	List \(- II\)
\((1)\) Rotation period of earth	\((i)\) \(10^5\, s\)
\((2)\) Revolution period of earth	\((ii)\) \(10^7\, s\)
\((3)\) Period of light wave	\((iii)\) \(10^{-15}\, s\)
\((4)\) Period of sound wave	\((iv)\) \(10^{-3}\, s\)

The distance between two consecutive points with phase difference of \(60^{\circ}\) in a wave of frequency \(500\,Hz\) is \(6.0\,m\). The velocity with which wave is traveling is \(.........km / s\)

In the following circuit, the battery has an emf of \(2\mathrm{V}\) and an internal resistance of \(\frac{2}{3}\  \Omega\). The power consumption in the entire circuit is _______ \(W\).

In a photoelectric experiment ultraviolet light of wavelength \(280\, {nm}\) is used with lithium cathode having work function \(\phi=2.5\, {eV}\). If the wavelength of incident light is switched to \(400\, {nm}\), find out the change in the stopping potential (in \({V}\)). \(\left({h}=6.63 \times 10^{-34}\, {J} \cdots\right.\), \(\left.{c}=3 \times 10^{8} \,{ms}^{-1}\right)\) 

When a metal surface is illuminated by light of wavelength \(\lambda\), the stopping potential is \(8 \mathrm{~V}\). When the same surface is illuminated by light of wavelength \(3 \lambda\), stopping potential is \(2 \mathrm{~V}\). The threshold wavelength for this surface is _______.

A circular disc of radius \(b\) has a hole of radius \(a\) at its centre (see figure). If the mass per unit area of the disc varies as \(\left( {\frac{{{\sigma _0}}}{r}} \right)\), then the radius of gyration of the disc about its axis passing through the centre is