A travelling microscope is used to determine the refractive index of a glass slab. If \(40\) divisions are there in \(1 \; cm\) on main scale and \(50\) Vernier scale divisions are equal to \(49\) main scale divisions, then least count of the travelling microscope is \(\dots \; \times 10^{-6} \; m\)

The electric field of a plane electromagnetic wave propagating along the \(x\) direction in vacuum is \(\overrightarrow{ E }= E _{0} \hat{ j } \cos (\omega t - kx )\). The magnetic field \(\overrightarrow{ B },\) at the moment \(t =0\) is :

The reading of pressure metre attached with a closed pipe is \(4.5 \times 10^4 \mathrm{~N} / \mathrm{m}^2\). On opening the valve, water starts flowing and the reading of pressure metre falls to \(2.0 \times 10^4 \mathrm{~N} / \mathrm{m}^2\). The velocity of water is found to be \(\sqrt{\mathrm{V}} \mathrm{m} / \mathrm{s}\). The value of \(\mathrm{V}\) is__________

The total current supplied to the circuit as shown in figure by the \(5 V\) battery is \(A\)

A liquid column of height \(0.04 \mathrm{~cm}\) balances excess pressure of soap bubble of certain radius. If density of liquid is \(8 \times 10^3 \mathrm{~kg} \mathrm{~m}^{-3}\) and surface tension of soap solution is \(0.28 \mathrm{Nm}^{-1}\), then diameter of the soap bubble is _______ \(\mathrm{cm}\). \(\text { (if } g=10 \mathrm{~ms}^{-2} \text { ) }\)

A particle of mass \(m\) is moving along the \(x\) -axis with initial velocity \(u \hat i\). It collides elastically with a particle of mass \(10\, m\) at rest and then moves with half its initial kinetic energy (see figure). If \(\sin \theta_{1}=\sqrt{n} \sin \theta_{2}\) then value of \(n\) is.....

A parallel plate capacitor of area \(60\, cm^2\) and separation \(3\, mm\) is charged initially to \(90\, \mu C\). If the medium between the plate gets slightly conducting and the plate loses the charge initially at the rate of \(2.5\times10^{-8}\, C/s\), then what is the magnetic field between the plates ?

A square shaped coil of area \(70\,cm ^2\) having \(600\) turns rotates in a magnetic field of \(0.4\,wbm ^{-2}\), about an axis which is parallel to one of the side of the coil and perpendicular to the direction of field. If the coil completes \(500\) revolution in a minute, the instantaneous emf when the plane of the coil is inclined at \(60^{\circ}\) with the field, will be \(..........V\). (Take \(\pi=\frac{22}{7}\) )

A uniformly thick wheel with moment of inertia \(I\) and radius \(R\) is free to rotate about its centre of mass (see fig). A massless string is wrapped over its rim and two blocks of masses \(\mathrm{m}_{1}\) and \(\mathrm{m}_{2}\left(\mathrm{m}_{1}>\mathrm{m}_{2}\right)\) are attached to the ends of the string. The system is released from rest. The angular speed of the wheel when \(\mathrm{m}_{1}\) descents by a distance \(h\) is

An infinitely long straight wire carrying current \(I\) is bent in a planer shape as shown in the diagram. The radius of the circular part is r . The magnetic field at the centre O of the circular loop is :

Some distant star is to be observed by some telescope of diameter of objective lens \(a\), at an angular resolution of \(3.0\times 10^{-7}\) radian. If the wavelength of light from the star reaching the telescope is \(500\) nm, the minimum diameter of the objective lens of the telescope is ________ cm. (nearest integer)