A paralle plate capacitor is made up of stair like structure with a palte area \(A\) of each stair and that is connected with a wire of length \(b\), as shown in the figure. The capacitance of the arrangement is \(\frac{ x }{15} \frac{\varepsilon_{0} A }{ b }\). The value of \(x\) is ............

A uniform cylinder of length \(L\) and mass \(M\) having crosssectional area \(A\) is suspended, with its length vertical, from a fixed point by a massless spring such that it is half submerged in a liquid of density \(\sigma\) at equilibrium position. The extension \(x_0\) of the spring when it is in equilibrium is

A student measuring the diameter of a pencil of circular cross-section with the help of a vernier scale records the following four readings \(5.50\, mm , 5.55\, mm ,\)\( 5.45\, mm ; 5.65\, mm .\) The average of these four readings is \(5.5375\, mm\) and the standard deviation of the data is \(0.07395\, mm .\) The average diameter of the pencil should therefore be recorded as

A charge particle moving in magnetic field \(B\), has the components of velocity along \(B\) as well as perpendicular to \(B\). The path of the charge particle will be

The space between the plates of a parallel plate capacitor is filled with a 'dielectric' whose 'dielectric constant' varies with distance as per the relation: \(K(x) = K_0 + \lambda x\) ( \(\lambda  =\) constant) The capacitance \(C,\) of the capacitor, would be related to its vacuum capacitance \(C_0\) for the relation

In the given circuit the value of \(\left|\frac{I_1+I_3}{I_2}\right|\) is

A roller is mad e by joining  together two cones at their vertices \(O\). It is kept on two rails \(AB\) and \(CD\), which are placed asymmetrically (see figure) , with its axis perpendicular to \(CD  \) and its centre \(O\) at the centre ofline joining \(AB\) and \(CD\) (see figure) . It is given a light push so that it \(A\) starts rolling with its centre \(O\) moving parallel to \(CD\) in the direction shown. As it moves, the roller will tend to

In a Young's double slit experiment, the width of the one of the slit is three times the other slit. The amplitude of the light coming from a slit is proportional to the slit-width. Find the ratio of the maximum to the minimum intensity in the interference pattern.

A moving coil galvanometer has \(100\) turns and each turn has an area of \(2.0 \mathrm{~cm}^2\). The magnetic field produced by the magnet is \(0.01 \mathrm{~T}\) and the deflection in the coil is \(0.05\) radian when a current of \(10 \mathrm{~mA}\) is passed through it. The torsional constant of the suspension wire is \(\mathrm{x} \times 10^{-5} \mathrm{~N}-\mathrm{m} / \mathrm{rad}\). The value of \(\mathrm{x}\) is _______.

Two identical symmetric double convex lenses of focal length \(f\) are cut into two equal parts \(L_1, L_2\) by \(A B\) plane and \(L_3, L_4\) by \(X Y\) plane as shown in figure respectively. The ratio of focal lengths of lenses \(L_1\) and \(L_3\) is ________.

A thin circular ring of mass \(M\) and radius \(R\) is rotating with a constant angular velocity \(2 \; rads ^{-1}\) in a horizontal plane about an axis vertical to its plane and passing through the center of the ring. If two objects each of mass \(m\) be attached gently to the opposite ends of a diameter of ring, the ring will then rotate with an angular velocity (in \(rads ^{-1}\) )

If momentum \((P),\) area \((A)\) and time \((T)\) are taken to be the fundamental quantities then the dimensional formula for energy is :