A current carrying is placed vertically and a particle of mass m with charge Q is released from rest. The particle moves along the axis of solenoid. If \(g\) is acceleration due to gravity then the acceleration (a) of the charged particle will satisfy :

An inductor of \(10\) mH, capacitor of \(0.1\ \mu\)F and a resistor of \(100\ \Omega\) are connected in series across an \(a.c\) power supply \(220\) V, \(70\) Hz. The power factor of the given circuit is \(0.5\). The difference in the inductive reactance and capacitance reactance is \(\sqrt{3}\alpha\ \Omega\). The value of \(\alpha\) is _____.

When a long glass capillary tube of radius \(0.015\;cm\) is dipped in a liquid, the liquid rises to a height of \(15\, cm\) within it. If the contact angle between the liquid and glass to clase to \(0^{\circ},\) the surface tension of the liquid, in \(milli\,Newton \;m ^{-1}\) is \(.....\) \(\left[\rho_{\text {(liquid) }}=900\; kg\,m ^{-3}, g =10\, ms ^{-2}\right]\) (Give answer in closest integer)

The one division of main scale of vernier callipers reads \(1\,mm\) and \(10\) divisions of Vernier scale is equal to the \(9\) divisions on main scale. When the two jaws of the instrument touch each other the \(zero\) of the Vernier lies to the right of \(zero\) of the main scale and its fourth division coincides with a main scale division. When a spherical bob is tightly placed between the two jaws, the \(zero\) of the Vernier scale lies in between \(4.1\,cm\) and \(4.2\,cm\) and \(6^{\text {th }}\) Vernier division coincides with a main scale division. The diameter of the bob will be \(.............10^{-2}\,cm\)

A uniform metal chain of mass \(m\) and length ' \(L\) ' passes over a massless and frictionless pulley. It is released from rest with a part of its length ' \(l\) ' is hanging on one side and rest of its length ' \(L -l\) ' is hanging on the other side of the pulley. At a certain point of time, when \(l=\frac{L}{x}\), the acceleration of the chain is \(\frac{g}{2}\). The value of \(x\) is ........

A satellite of \(10^3 \mathrm{~kg}\) mass is revolving in circular orbit of radius \(2 \mathrm{R}\). If \(\frac{10^4 \mathrm{R}}{6} \mathrm{~J}\) energy is supplied to the satellite, it would revolve in a new circular orbit of radius _______.
(use \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2, \mathrm{R}=\) radius of earth)

The circuit contains two diodes each with a forward resistance of \(50\, \Omega\) and with infinite reverse resistance. If the battery voltage is \(6\, V\), the current through the \(120\, \Omega\) resistance is \(mA\)

A capacitor has capacitance \(5 \mu F\) when it's parallel plates are separated by air medium of thickness \(d\). A slab of material of dielectric constant \(1.5\) having area equal to that of plates but thickness \(\frac{ d }{2}\) is inserted between the plates. Capacitance of the capacitor in the presence of slab will be \(..........\mu F\)

Water flows into a large tank with flat bottom at the rate of \({10^{ - 4}}\,{m^3}{s^{ - 1}}\). Water is also leaking out of a hole of area \(1\, cm^2\) at its bottom. If the height of the water in the tank remains steady, then this height is........ \(cm\)

Calculate the value of mean free path \((\lambda)\) for oxygen molecules at temperature \(27^{\circ}\, C\) and pressure \(1.01 \times 10^{5} \,Pa\). Assume the molecular diameter \(0.3 \,nm\) and the gas is ideal. \(\left( k =1.38 \times 10^{-23}\, J\,K ^{-1}\right)\) (in \(nm\))

The equivalent resistance between \(A\) and \(B\) of the network shown in figure

If \(x^2 + y^2 + sin\, y = 4\), then the value of \(\frac{{{d^2}y}}{{d{x^2}}}\) at the point \((- 2, 0)\) is

If the number of integral terms in the expansion of \(\left(3^{\frac{1}{2}}+5^{\frac{1}{8}}\right)^{\text {n }}\) is exactly \(33,\) then the least value of \(n\) is