A uniform heating wire of resistance \(36\, \Omega\) is connected across a potential difference of \(240\, {V}\) The wire is then cut into half and potential difference of \(240\, {V}\) is applied across each half separately. The ratio of power dissipation in first case to the total power dissipation in the second case would be \(1: {x}\), where \({x}\) is........... .

A body cools from \(80^{\circ}\,C\) to \(60^{\circ}\,C\) in \(5\) minutes. The temperature of the surrounding is \(20^{\circ} C\). The time it takes to cool from \(60^{\circ}\,C\) to \(40^{\circ}\,C\) is........... \(s\)

Two wires \(W_1\) and \(W_2\) have the same radius \(r\) and respective densities \({\rho _1}\) and \({\rho _2}\) such that \({\rho _2} = 4{\rho _1}\). They are joined together at the point \(O\), as  shown in the figure. The combination is used as a sonometer wire and kept under tension \(T\). The point \(O\) is midway between the two bridges. When a stationary waves is set up in the composite wire, the joint is found to be a node. The ratio of the number of an tin odes formed in \(W_1\) to \(W_2\) is

A capacitor of capacitance \(50 \; pF\) is charged by \(100 \; V\) source. It is then connected to another uncharged identical capacitor. Electrostatic energy loss in the process is \(\dots \; nJ\).

Two moles of an ideal gas with \(\frac{\mathrm{C}_{\mathrm{P}}}{\mathrm{C}_{\mathrm{V}}}=\frac{5}{3}\) are mixed with \(3\) moles of another ideal gas with \(\frac{\mathrm{C}_{\mathrm{P}}}{\mathrm{C}_{\mathrm{V}}}=\frac{4}{3} .\) The value of \(\frac{\mathrm{C}_{\mathrm{P}}}{\mathrm{C}_{\mathrm{V}}}\) for the mixture is

A \(100 \,\Omega\) resistance, a \(0.1\, \mu {F}\) capacitor and an inductor are connected in series across a \(250 \,{V}\) supply at variable frequeny. Calculate the value of inductance of inductor at which resonance will occur. Given that the resonant frequency is \(60\, {Hz}\). (In \({H}\))

A physical quantity \(p\) is described by the relation \(p\, = a^{1/2}\, b^2\, c^3\, d^{-4}\) If the relative errors in the measurement of \(a, b, c\) and \(d\) respectively, are \(2\% , 1\%, 3\%\) and \(5\%\), then the relative error in \(P\) will be ........... \(\%\)

Two polarisers \(P_1\) and \(P_2\) are placed in such a way that the intensity of the transmitted light will be zero. A third polariser \(P_3\) is inserted in between \(P_1\) and \(\mathrm{P}_2\), at the particular angle between \(\mathrm{P}_2\) and \(\mathrm{P}_3\). The transmitted intensity of the light passing the through all three polarisers is maximum. The angle between the polarisers \(\mathrm{P}_2\) and \(\mathrm{P}_3\) is :

A \(30\) cm long solenoid has \(10\) turns per cm and area of \(5\) cm\(^2\). The current through the solenoid coil varies from \(2\) A to \(4\) A in \(3.14\) s. The e.m.f. induced in the coil is \(\alpha\times 10^{-5}\) V. The value \(\alpha\) is ________.