A parallel beam of light travelling in air (refractive index 1.0) is incident on a convex spherical glass surface of radius of curvature 50 cm . Refractive index of glass is 1.5 . The rays converge to a point at a distance x cm from the centre of the curvature of the spherical surface. The value of x is __________ cm.

A small particle moves to position \(5 \hat{i}-2 \hat{j}+\hat{k}\) from its initial position \(2 \hat{i}+3 \hat{j}-4 \hat{k}\) under the action of force \(5 \hat{i}+2 \hat{j}+7 \hat{k} N\). The value of work done will be \(............J\).

A small square loop of side \('a'\) and one turn is placed inside a larger square loop of side \({b}\) and one turn \((b \gg a)\). The two loops are coplanar with thei centres coinciding. If a current \(I\) is passed in the square loop of side \('b',\) then the coefficient of mutual inductance between the two loops is

Two trains ' \(A\) ' and ' \(B\) ' of length ' \(l\) ' and ' \(4 l\) ' are travelling into a tunnel of length ' \(L\) ' in parallel tracks from opposite directions with velocities \(108\,km / h\) and \(72\,km / h\), respectively. If train '\(A\)' takes \(35\,s\) less time than train ' \(B\) ' to cross the tunnel then, length ' \(L\) ' of tunnel is \(...........\,m\) \(\text { (Given } L=60 l \text { ) }\)

A coil of cross-sectional area  \(A\) having \(n\)  turns is placed in a uniform magnetic field \(B.\)  When it is rotated with an angular velocity \(\omega ,\)  the maximum \(e.m.f.\)  induced in the coil will be

A planar loop made of wire is rotating in a uniform magnetic field. At time \(t=0\), the plane of the loop is perpendicular to the magnetic field. If the loop is rotating about an axis passing through its plane with a period of \(10\) seconds, at which of the following times will the induced electromotive force (emf) be maximum and minimum, respectively?

A wire carrying current \(I\) is bent in the shape \(A\,B\,C\,D\,E\,F\,A\) as shown, where rectangle \(A\,B\,C\,D\,A\) and \(A\,D\,E\,F\,A\) are perpendicular to each other. If the sides of the rectangles are of lengths \(a\) and \(b,\) then the magnitude and direction of magnetic moment of the loop \(A\,B\,C\,D\,E\,F\,A\,\) is

Two infinite planes each with uniform surface charge density \(+\sigma\) are kept in such a way that the angle between them is \(30^{\circ} .\) The electric field in the region shown between them is given by

Starting from the origin at time \(t=0,\) with initial velocity \(5 \hat{ j }\, ms ^{-1},\) a particle moves in the \(x-y\) plane with a constant acceleration of \((10 \hat{ i }+4 \hat{ j })\, ms ^{-2}\). At time \(t\), its coordinates are \(\left(20\, m , y _{0}\, m \right) .\) The values of \(t\) and \(y _{0},\) are respectively

Three charges +2q. +3q and -4q are situated at (0,-3a), (2a, 0) and (-2a,0) respectively in the xy plane. The resultant dipole moment about origin is _________

A metal rod of length \(L\) rotates about one end at origin with a uniform angular velocity \(\omega\). The magnetic field radially falls off as \(B(r) = B_0 e^{-\lambda r}\); \(\lambda\) being a positive constant. The emf induced (neglecting the centripetal force on electrons in the rod) is :

\(0.056 \,kg\) of Nitrogen is enclosed in a vessel at a temperature of \(127\,^{\circ} C\). The amount of heat required to double the speed of its molecules is k cal. (Take \(R =2\) \(cal \,mole\) \(^{-1} K ^{-1}\) )