Consider a force \(\overrightarrow{\mathrm{F}}=-\mathrm{x} \hat{\mathrm{i}}+\mathrm{y} \hat{\mathrm{j}}\). The work done by this force in moving a particle from point \(\mathrm{A}(1,0)\) to \(\mathrm{B}(0,1)\) along the line segment is  (all quantities are in \(SI\) units)

An AC voltage \(\mathrm{V}=20 \sin 200 \pi t\) is applied to a series \(LCR\) circuit which drives a current \(I=10 \sin \left(200 \pi t+\frac{\pi}{3}\right)\). The average power dissipated is _______.

The magnetic intensity at the centre of a long current carrying solenoid is found to be \(1.6 \times 10^3 Am ^{-1}\). If the number of turns is 8 per cm, then the current flowing through the solenoid is \(................\,A\)

A proton, an electron and an alpha particle have the same energies. Their de-Broglie wavelengths will be compared as _______.

The magnetic field in a region is given by \(\overrightarrow{ B }= B _{0}\left(\frac{ x }{ a }\right) \,\hat{ k }\). A square loop of side \(d\) is placed with its edges along the \(x\) and \(y\) axes. The loop is moved with a constant velocity \(\overrightarrow{ v }= v _{0} \hat{ i }\). The emf induced in the loop is:

The correct relation between \(\gamma=\frac{ C _p}{ c _v}\) and temperature \(T\) is :

As shown in the diagram, when the incident ray is parallel to base of the prism, the emergent ray grazes along the second surface.

If refractive index of the material of prism is \(\sqrt{2}\), the angle \(\theta\) of prism is.

A uniformly tapering conical wire is made from a material of Young's modulus \(Y\)  and has a normal, unextended length \(L.\) The radii, at the upper and lower ends of this conical wire, have values \(R\) and \(3R,\)  respectively. The upper end of the wire is fixed to a rigid support and a mass \(M\) is suspended from its lower end. The equilibrium extended length, of this wire, would equal 

In a Young's double slit experiment, the intensity at some point on the screen is found to be \(\dfrac{3}{4}\) times of the maximum of the interference pattern. The path difference between the interfering waves at this point is \(\dfrac{\lambda}{x}\) where \(\lambda\) is wavelength of the incident light. The value of \(x\) is _______.

Given below are two statements: Statement \(I:\) Area under velocity- time graph gives the distance travelled by the body in a given time. Statement \(II:\) Area under acceleration- time graph is equal to the change in velocity- in the given time. In the light of given statements, choose the correct answer from the options given below.

Two metal spheres of radius \(R\) and \(3 R\) have same surface charge density \(\sigma\). If they are brought in contact and then separated, the surface charge density on smaller and bigger sphere becomes \(\sigma_1\) and \(\sigma_2\), respectively. The ratio \(\frac{\sigma_1}{\sigma_2}\) is ________.