A single slit of width \(a\) is illuminated by a monochromatic light of wavelength \(600\,nm\). The value of \(a\) for which first minimum appears at \(\theta=30^{\circ}\) on the screen will be ........... \(\mu m\)

If \(\varepsilon_0\) is the permittivity of free space and \(E\) is the electric field, then \(\varepsilon_0 E^2\) has the dimensions is _______.

Two coherent point sources \(S_1\) and \(S_2\) are separated by a small distance '\(d\)' as shown. The fringes obtained on the screen will be

An ideal fluid flows (laminar flow) through a pipe of non-uniform diameter. The maximum and minimum diameters of the pipes are \(6.4 \;\mathrm{cm}\) and \(4.8 \;\mathrm{cm},\) respectively. The ratio of the minimum and the maximum velocities of fluid in this pipe is:

Match List - I with List - II.

List - I	List - II
(A) Young’s Modulus	(I) \(M L^{-1} T^{-1}\)
(B) Torque	(II) \(M L^{-1} T^{-2}\)
(C) Coefficient of Viscosity	(III) \(M^{-1} L^3 T^{-2}\)
(D) Gravitational Constant	(IV) \(M L^2 T^{-2}\)

Choose the correct answer from the options given below :

Two identical positive charges \(Q\) each are fixed at a distance of ' \(2 a\) ' apart from each other. Another point charge qo with mass ' \(m\) ' is placed at midpoint between two fixed charges. For a small displacement along the line joining the fixed charges, the charge \(q_{0}\) executes \(SHM\). The time period of oscillation of charge \(q_{0}\) will be.

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.

The deflection in moving coil galvanometer falls from \(25\) divisions to \(5\) division when a shunt of \(24\  \Omega\) is applied. The resistance of galvanometer coil will be :

Assume that a tunnel is dug along a chord of the earth, at a perpendicular distance \(( R / 2)\) from the earth's centre, where \('R'\) is the radius of the Earth. The wall of the tunnel is frictionless. If a particle is released in this tunnel, it will execute a simple harmonic motion with a time period

In the figure given, the electric current flowing through the \(5\, k \Omega\) resistor is \('x' mA.\) The value of \(x\) to the nearest integer is .........

A sand dropper drops sand of mass \(m(t)\) on a conveyer belt at a rate proportional to the square root of speed \((v)\) of the belt, i.e. \(\frac{\mathrm{dm}}{\mathrm{dt}} \propto \sqrt{v}\). If P is the power delivered to run the belt at constant speed then which of the following relationship is true?