A small bob of mass 100 mg and charge \(+10 \mu \mathrm{C}\) is connected to an insulating string of length 1 m. It is brought near to an infinitely long nonconducting sheet of charge density ' \(\sigma\) ' as shown in figure. If string subtends an angle of \(45^{\circ}\) with the sheet at equilibrium the charge density of sheet will be :
(Given, \(\varepsilon_0=8.85 \times 10^{-12} \frac{\mathrm{~F}}{\mathrm{~m}}\) and acceleration due to gravity, \(g=10 \mathrm{~m} / \mathrm{s}^2\))

\(c_P\) and \(c_V\) are specific heats at constant pressure and constant volume respectively. It is observed that \(c_P - c_V = a\) for hydrogen gas \(c_P - c_V = b\) for nitrogen gas The correct relation between \(a\) and \(b\) is

In a photoelectric effect experiment a light of frequency \(1.5\) times the threshold frequency is made to fall on the surface of photosensitive material. Now if the frequency is halved and intensity is doubled, the number of photo electrons emitted will be _______.

A positive ion \(A\) and a negative ion \(B\) has charges \(6.67 \times 10^{-19} \mathrm{C}\) and \(9.6 \times 10^{-10} \mathrm{C}\), and masses \(19.2 \times 10^{-27} \mathrm{~kg}\) and \(9 \times 10^{-27} \mathrm{~kg}\) respectively. At an instant, the ions are separated by a certain distance r. At that instant the ratio of the magnitudes of electrostatic force to gravitational force is \(P \times 10^{45}\), where the value of 10\(P\) is _______.
(Take \(\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \mathrm{Nm}^2 \mathrm{C}^{-1}\) and universal gravitational constant as \(6.67 \times 10^{-11} \mathrm{Nm}^2 \mathrm{~kg}^{-2}\) )
Assume that charge may not be an integral multiple of electrons.

Consider a planet in some solar system which has a mass double the mass of earth and density equal to the average density of earth. If the weight of an object on earth is \({W}\), then weight of the same object on that planet will be

In a series \(LCR\) circuit, the inductive reactance \(\left( X _{ L }\right)\) is \(10\, \Omega\) and the capacitive reactance \(\left( X _{ C }\right)\) is \(4\, \Omega\). The resistance \(( R )\) in the circuit is \(6\, \Omega\). The power factor of the circuit is :

At very high frequencies, the effective impendance of the given circuit will be \(....\Omega\)

The lines \(\mathrm{L}_1, \mathrm{~L}_2, \ldots, \mathrm{L}_{20}\) are distinct. For \(\mathrm{n}=1,2,3, \ldots, 10\) all the lines \(\mathrm{L}_{2 \mathrm{n}-1}\) are parallel to each other and all the lines \(L_{2 n}\) pass through a given point \(P\). The maximum number of points of intersection of pairs of lines from the set \(\left\{\mathrm{L}_1, \mathrm{~L}_2, \ldots, \mathrm{L}_{20}\right\}\) is equal to :

Let \(R\) be a relation defined on \(N\) as a \(R\) b is \(2 a+3 b\) is a multiple of \(5, a, b \in N\). Then \(R\) is

Let \(p\) and \(q\) be two real numbers such that \(p+q=\) 3 and \(p^{4}+q^{4}=369\). Then \(\left(\frac{1}{p}+\frac{1}{q}\right)^{-2}\) is equal to

In an experiment with photoelectric effect, the stopping potential,

For a specific wavelength \(670 \; nm\) of light coming from a galaxy moving with velocity \(v\), the observed wavelength is \(670.7 \; nm\). The value of \(v\) is ............

The \(8^{\text {th }}\) common term of the series \(S _1=3+7+11+15+19+\ldots . .\) ; \(S _2=1+6+11+16+21+\ldots .\) is \(.......\).