A particle experiences a variable force \(\overrightarrow{ F }=\left(4 x \hat{ i }+3 y ^{2} \hat{ j }\right)\) in a horizontal \(x - y\) plane. Assume distance in meters and force is newton. If the particle moves from point \((1,2)\) to point \((2,3)\) in the \(x-y\) plane, the Kinetic Energy changes by............\(j\)

A proton with a kinetic energy of \(2.0\,eV\) moves into a region of uniform magnetic field of magnitude \(\frac{\pi}{2} \times 10^{-3}\,T\). The angle between the direction of magnetic field and velocity of proton is \(60^{\circ}\). The pitch of the helical path taken by the proton is \(..........cm\) (Take, mass of proton \(=1.6 \times 10^{-27}\,kg\) and Charge on proton \(=1.6 \times 10^{-19}\,kg)\)

A parallel plate capacitor filled with a medium of dielectric constant \(10\) , is connected across a battery and is charged. The dielectric slab is replaced by another slab of dielectric constant \(15\) . Then the energy of capacitor will ......................

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,

Two long current carrying conductors are placed parallel to each other at a distance of \(8 \,cm\) between them. The magnitude of magnetic field produced at mid-point between the two conductors due to current flowing in them is \(300 \,\mu T\). The equal current flowing in the two conductors is ...............

When photons of wavelength \(\lambda _1\)  are incident on an isolated sphere, the corresponding stopping potential is found to be \(V.\)  When photons of wavelength \(\lambda _2\) are used, the corresponding stopping potential was thrice that of the above value . If light of wavelength \(\lambda _3\) is used then find the stopping potential for this case

A particle moves from the point \(\left( {2.0\hat i + 4.0\hat j} \right)\,m\), at \(t = 0\) with an initial velocity \(\left( {5.0\hat i + 4.0\hat j} \right)\,m{s^{ - 1  }}\). It is acted upon by a constant force which produces a constant acceleration \(\left( {4.0\hat i + 4.0\hat j} \right)\,m{s^{ - 2}}\). What is the distance of the particle from the origin at time \(2\,s\)

A uniform cylinder of length \(L\) and mass \(M\) having crosssectional area \(A\) is suspended, with its length vertical, from a fixed point by a massless spring such that it is half submerged in a liquid of density \(\sigma\) at equilibrium position. The extension \(x_0\) of the spring when it is in equilibrium is

A circuit element \(X\) when connected to an a.c. supply of peak voltage \(100\,V\) gives a peak current of \(5\,A\) which is in phase with the voltage. A second element \(Y\) when connected to the same a.c. supply also gives the same value of peak current which lags behind the voltage by \(\frac{\pi}{2}\). If \(X\) and \(Y\) are connected in series to the same supply, what will be the rms value of the current in ampere?

A body of mass \(2\,kg\) is initially at rest. It starts moving unidirectionally under the influence of a source of constant 10.Power P. Its displacement in \(4\,s\) is \(\frac{1}{3} \alpha^2 \sqrt{ P } m\). The value of \(\alpha\) will be \(.............\)

Monochromatic light of wavelength \(500 \mathrm{~nm}\) is used in Young's double slit experiment. An interference pattern is obtained on a screen When one of the slits is covered with a very thin glass plate (refractive index \(=1.5\) ), the central maximum is shifted to a position previously occupied by the \(4^{\text {th }}\) bright fringe. The thickness of the glass-plate is _______\(\mu \mathrm{m}\).

A heater is designed to operate with a power of \(1000 \mathrm{~W}\) in a \(100 \mathrm{~V}\) line. It is connected in combination with a resistance of \(10 \Omega\) and a resistance \(R\), to a \(100 \mathrm{~V}\) mains as shown in figure. For the heater to operate at \(62.5 \mathrm{~W}\), the value of \(\mathrm{R}\) should be _______ \(\Omega\).