The current density in a cylindrical wire of radius \(4 \; mm\) is \(4 \times 10^{6} \; Am ^{-2}\). The current through the outer portion of the wire between radial distance \(\frac{R}{2}\) and \(R\) is \(\dots \; \pi A .\)

The moment of inertia of a uniform cylinder of length \(l\) and radius \(R\) about its perpendicular bisector is \(I\). What is the ratio \(l/R\) such that the moment of inertia is minimum?

Two planets \(A\) and \(B\) of radii \(R\) and \(1.5 R\) have densities \(\rho\) and \(\rho / 2\) respectively. The ratio of acceleration due to gravity at the surface of \(B\) to \(A\) is :

The size of the images of an object, formed by a thin lens are equal when the object is placed at two different positions 8 cm and 24 cm from the lens. The focal length of the lens is ___________ cm.

The potential energy function (in J) of a particle in a region of space is given as \(U=\left(2 x^2+3 y^3+2 z\right)\). Here \(\mathrm{x}, \mathrm{y}\) and \(\mathrm{z}\) are in meter. The magnitude of \(\mathrm{x}\) - component of force (in \(\mathrm{N}\) ) acting on the particle at point \(\mathrm{P}(1,2,3)\ \mathrm{m}\) is:

Voltage rating of a parallel plate capacitor is \(500\,V\). Its dielectric can withstand a maximum electric field of \({10^6}\,\frac{V}{m}\). The plate area is \(10^{-4}\, m^2\) . What is the dielectric constant if the capacitance is \(15\, pF\) ? (given \({ \in _0} = 8.86 \times {10^{ - 12}}\,{C^2}\,/N{m^2}\))

Given below are two statements: Statement \(I:\) If \(E\) be the total energy of a satellite moving around the earth, then its potential energy will be \(\frac{E}{2}\). Statement \(II:\) The kinetic energy of a satellite revolving in an orbit is equal to the half the magnitude of total energy \(E\). In the light of the above statements, choose the most appropriate answer from the options given below

In a double -slit experiment, green light \(\left( {5303\,\mathop A\limits^o } \right)\) falls on a double slit having a separation of \(19.44\,\mu m\) and a width of \(4.05\,\mu m\). The number of bright fringes between the first and the second diffraction minima is

Given below is the plot of a potential energy function \(U(x)\) for a system, in which a particle is in one dimensional motion, while a conservative force \(F(x)\) acts on it. Suppose that \(E_{\text {mech }}=8\, {J}\), the incorrect statement for this system is:

A particle is travelling \(4\) times as fast as an electron. Assuming the ratio of de-Broglie wavelength of a particle to that of electron is \(2: 1,\) the mass of the particle is

A wire of length 10 cm and diameter 0.5 mm is used in a bulb. The temperature of the wire is \(1727^{\circ} \mathrm{C}\) and power radiated by the wire is 94.2 W. Its emissivity is \(\frac{x}{8}\) where \(x=\)_______
(Given \(\sigma=6.0 \times 10^{-8} \mathrm{~W} \mathrm{~m}^{-2} \mathrm{~K}^{-4}, \pi=3.14\) and assume that the emissivity of wire material is same at all wavelength.)

At the centre of a fixed large circular coil of radius \(R\), a much smaller circular coil of radius \(r\) is placed. The two coils are concentric and are in the same plane. The larger coil carries a current \(I\). The smaller coil is set to rotate with a constant angular velocity \(\omega \) about an axis along their common diameter. Calculate the \(emf\) induced in the smaller coil after a time \(t\) of its start of rotation