A circular coil having \(N\,turns\) and radius \(r\) carries a current. It is held in the \(XZ\) plane in a magnetic field \(B\hat i\) . The torque on the coil due to the magnetic field is

Calculate the amount of charge on capacitor of \(4\, \mu\, {F}\). The internal resistance of battery is \(1\, \Omega\) : (in \(\mu {C}\))

The magnetic field vector of an electromagnetic wave is given by \({B}={B}_{o} \frac{\hat{{i}}+\hat{{j}}}{\sqrt{2}} \cos ({kz}-\omega {t})\); where \(\hat{i}, \hat{j}\) represents unit vector along \({x}\) and \({y}\)-axis respectively. At \(t=0\, {s}\), two electric charges \(q_{1}\) of \(4\, \pi\) coulomb and \({q}_{2}\) of \(2 \,\pi\) coulomb located at \(\left(0,0, \frac{\pi}{{k}}\right)\) and \(\left(0,0, \frac{3 \pi}{{k}}\right)\), respectively, have the same velocity of \(0.5 \,{c} \hat{{i}}\), (where \({c}\) is the velocity of light). The ratio of the force acting on charge \({q}_{1}\) to \({q}_{2}\) is :-

The mass of proton, neutron and helium nucleus are respectively \(1.0073\,u , 1.0087\,u\) and \(4.0015\,u\). The binding energy of helium nucleus is \(.........\,MeV\)

A one metre long (both ends open) organ pipe is kept in a gas that has double the density of air at \(STP\). Assuming the speed of sound in air at \(STP\) is \(300\; \mathrm{m} / \mathrm{s}\), the frequency difference between the fundamental and second harmonic of this pipe is ___ \(\mathrm{Hz}\)

A circular plate is rotating in horizontal plane, about an axis passing through its center and perpendicular to the plate, with an angular velocity \(\omega\). A person sits at the center having two dumbbells in his hands. When he stretches out his hands, the moment of inertia of the system becomes triple. If \(E\) be the initial Kinetic energy of the system, then final Kinetic energy will be \(\frac{E}{x}\).The value of \(x\) is \(....\)

A particle moving in a straight line covers half the distance with speed \(6 \mathrm{~m} / \mathrm{s}\). The other half is covered in two equal time intervals with speeds 9 \(\mathrm{m} / \mathrm{s}\) and \(15 \mathrm{~m} / \mathrm{s}\) respectively. The average speed of the particle during the motion is _______.

A horse rider covers half the distance with \(5\,m / s\) speed. The remaining part of the distance was travelled with speed \(10\,m / s\) for half the time and with speed \(15 m / s\) for other half of the time. The mean speed of the rider averaged over the whole time of motion is \(\frac{x}{7}\,m / s\). The value of \(x\) is

Let each of the two ellipses \(E _1: \frac{ x ^2}{ a ^2}+\frac{ y ^2}{b^2}=1,( a > b )\) and \(E _2: \frac{ x ^2}{A^2}+\frac{ y ^2}{B^2}=1,(A< B )\) have eccentricity \(\frac{4}{5}\). Let the lengths of the latus recta of \(E_1\) and \(E_2\) be \(\ell_1\) and \(\ell_2\), respectively, such that \(2 \ell_1^2=9 \ell_2\). If the distance between the foci of \(E_1\) is 8 , then the distance between the foci of \(E _2\) is

Let us consider a curve, \(\mathrm{y}=\mathrm{f}(\mathrm{x})\) passing through the point \((-2,2)\) and the slope of the tangent to the curve at any point \((x, f(x))\) is given by \(f(x)+x f^{\prime}(x)=x^{2}\) Then :

The figure shows two solid discs with radius \({R}\) and \({r}\) respectively. If mass per unit area is same for both, what is the ratio of \({MI}\) of bigger disc around axis \({AB}\) (Which is \(\perp\) to the plane of the disc and passing through its centre) of MI of smaller disc around one of its diameters lying on its plane? Given \('M'\) is the mass of the larger disc. (\(MI\) stands for moment of inertia)

The coefficient of \(x ^7\) in \(\left(1-x+2 x^3\right)^{10}\) is \(........\).

If the line \(\frac{{x - 3}}{2} = \frac{{y + 2}}{{ - 1}} = \frac{{z + 4}}{3}\) lies in the plane  \(lx + my - z = 9\) then \({l^2} + {m^2} = \;.\;.\;.\;.\;.\;\)