A potentiometer wire \(AB\) having length \(L\) and resistance \(12\, r\) is joined to a cell \(D\) of \(emf\) \(\varepsilon \) and internal resistance \(r\). A cell \(C\) having \(emf\) \(\varepsilon /2\) and internal resistance \(3r\) is connected. The length \(AJ\) at which the galvanometer as shown in figure shows no deflection is

A Carnot engine take \(5000 \,k\,cal\) of heat from a reservoir at \(727\,^{\circ}C\) and gives heat to a \(\operatorname{sink}\) at \(127\,^{\circ}C\). The work done by the engine is \(.......... \times 10^{6}\,J\)

A ball of mass \(10\, kg\) moving with a velocity \(10 \sqrt{3} m / s\) along the \(x-\)axis, hits another ball of mass \(20\, kg\) which is at rest. After the collision, first ball comes to rest while the second ball disintegrates into two equal pieces. One piece starts moving along \(y-\)axis with a speed of \(10\) \(m / s\). The second piece starts moving at an angle of \(30^{\circ}\) with respect to the \(x-\)axis. The velocity of the ball moving at \(30^{\circ}\) with \(x-\) axis is \(x\, m / s\). The configuration of pieces after collision is shown in the figure below. The value of \(x\) to the nearest integer is

A moving coil galvanometer, having a resistance \(G\), produces full scale deflection when a current \(I_g\) flows through it. This galvanometer can be converted into \((i)\) an ammeter of range \(0\) to \(I_0 (I_0 > I_g)\) by connecting a shunt resistance \(R_A\) to it and \((ii)\) into a voltmeter of range \(0\) to \(V(V = GI_0)\) by connecting a series resistance \(R_V\) to it. Then,

The oscillating magnetic field in a plane electromagnetic wave is given by \(B _{ y }=5 \times 10^{-6} \sin\) \(1000\,\pi\left(5 x -4 \times 10^{8} t \right) T\). The amplitude of electric field will be.

Two strings with circular cross section and made of same material, are stretched to have same amount of tension. A transverse wave is then made to pass through both the strings. The velocity of the wave in the first string having the radius of cross section R is \(\mathrm{v}_1\), and that in the other string having radius of cross section \(R / 2\) is \(v_2\). Then \(\frac{v_2}{v_1}=\) ________.

Given below are two statements :
Statement (I) : The dimensions of Planck's constant and angular momentum are same.
Statement (II) : In Bohr's model electron revolve around the nucleus only in those orbits for which angular momentum is integral multiple of Planck's constant.
In the light of the above statements, choose the most appropriate answer from the options given below :

Identify the operation performed by the circuit given below

Both the nucleus and the atom of some element are in their respective first excited states. They get de-excited by emitting photons of wavelengths \(\lambda _N,\,\lambda _A\)  respectively. The ratio \(\frac{{{\lambda _N}}}{{{\lambda _A}}}\)  is closest to