A long cylindrical volume contains a uniformly distributed charge of density \(\rho \;Cm ^{-3}\). The electric field inside the cylindrical volume at a distance \(x =\frac{2 \varepsilon_{0}}{\rho} m\) from its axis is \(.......Vm ^{-1}\)

In the reported figure, a capacitor is formed by placing a compound dielectric between the plates of parallel plate capacitor. The expression for the capacity of the said capacitor will be (Given area of plate \(=A\) )

The displacement current of \(4.425 \,\mu A\) is developed in the space between the plates of parallel plate capacitor when voltage is changing at a rate of \(10^{6} \,Vs ^{-1}\). The area of each plate of the capacitor is \(40 \,cm ^{2}\). The distance between each plate of the capacitor is \(x \times 10^{-3} \,m\). The value of \(x\) is ................ (Permittivity of free space, \(\varepsilon _{0}=8.85 \times 10^{-12} \,C ^{2} N ^{-1} m ^{-2}\) )

An engine takes in \(5\) moles of air at \(20\,^{\circ} C\) and \(1\) \(atm,\) and compresses it adiabaticaly to \(1 / 10^{\text {th }}\) of the original volume. Assuming air to be a diatomic ideal gas made up of rigid molecules, the change in its internal energy during this process comes out to be \(X\, kJ\). The value of \(X\) to the nearest integer is

A block '\(A\)' takes \(2\,s\) to slide down a frictionless incline of \(30^{\circ}\) and length ' \(l\) ', kept inside a lift going up with uniform velocity ' \(v\) '. If the incline is changed to \(45^{\circ}\), the time taken by the block, to slide down the incline, will be approximately \(........\,s\)

A parallel plate air capacitor is connected to a battery. The plates are pulled apart at uniform speed \(v\). If \(x\) is the separation between the plates at any instant, then the time rate of change of electrostatic energy of the capacitor is proportional to \(x^\alpha\), where \(\alpha\) is _____.

A Carnot engine absorbs \(1000\,J\) of heat energy from a reservoir at \(127\,^oC\) and rejects \(600\,J\) of heat energy during each cycle. The efficiency of engine and temperature of sink will be

A wire of length 25 m and cross-sectional area \(5 \mathrm{~mm}^2\) having resistivity of \(2 \times 10^{-6} \Omega \mathrm{~m}\) is bent into a complete circle. The resistance between diametrically opposite points will be:

A body is in simple harmonic motion with time period half second \((T\, = 0.5\, s)\) and amplitude one \(cm\, (A\,= 1\, cm)\). Find the average velocity in the interval in which it moves form equilibrium position to half of its amplitude .... \(cm/s\)

Identify the characteristics of an adiabatic process in a monoatomic gas.
(A) Internal energy is constant.
(B) Work done in the process is equal to the charge in internal energy.
(C) The product of temperature and volume is a constant.
(D) The product of pressure and volume is a constant.
(E) The work done to change the temperature from \(\mathrm{T}_1\) to \(\mathrm{T}_2\) is proportional to \(\left(\mathrm{T}_2-\mathrm{T}_1\right)\)
Choose the correct answer from the options given below :

The dimensional formula of angular impulse is _______.