A proton and an \(\alpha -\) particle (with their masses in the ratio of \(1 : 4\) and charges in the ratio of \(1:2\) are accelerated from rest through a potential difference \(V\). If a uniform magnetic field \((B)\) is set up perpendicular to their velocities, the ratio of the radii \(r_p : r_{\alpha }\) of the circular paths described by them will be

A particle is oscillating on the \(X-\) axis with an amplitude \(2\, cm\) about the point \(x_0 = 10\, cm\) with a frequency \(\omega \). A concave mirror of focal length \(5\,
cm\) is placed at the origin ( see figure) Identify the correct statements \((A)\) The image executes periodic motion \((B)\) The image executes non-periodic motion \((C)\) The turning points of the image are asymmetric w.r.t the image of the point at \(x =10\,cm\) \((D)\) The distance between the turning points of the oscillation of the image is \(\frac{{100}}{{21}}\)

A hydrogen atom changes its state from \(n=3\) to \(\mathrm{n}=2\). Due to recoil, the percentage change in the wave length of emitted light is approximately \(1 \times 10^{-\mathrm{n}}\). The value of \(\mathrm{n}\) is _______. [Given \(\mathrm{Rhc}=13.6 \mathrm{eV}, \mathrm{hc}=1242 \mathrm{eV} \mathrm{nm}\), \(\mathrm{h}=6.6 \times 10^{-34} \mathrm{~J} \mathrm{~s}\), mass of the hydrogen atom \(\left.=1.6 \times 10^{-27} \mathrm{~kg}\right]\)

If \(B\) is magnetic field and \(\mu_0\) is permeability of free space, then the dimensions of \(\left(B / \mu_0\right)\) is

Two transparent media having refractive indices \(1.0\) and \(1.5\) are separated by a spherical refracting surface of radius of curvature \(30\,cm\). The centre of curvature of surface is towards denser medium and a point object is placed on the principle axis in rarer medium at a distance of \(15\,cm\) from the pole of the surface. The distance of image from the pole of the surface is .......\(cm\)

A big drop is formed by coalescing \(1000\) small droplets of water. The surface energy will become _______.

In a Young’s double slit experiment, the slits are placed \(0.320\,mm\) apart. Light of wavelength \(\lambda = 500\,nm\) is incident on the slits. The total number of bright fringes that are observed in the angular range \( - {30^o} \le \theta  \le {30^o}\) is

A hollow spherical shell at outer radius \(R\) floats just submerged under the water surface. The inner radius of the shell is \(r\). If the specific gravity of the shell material is \(\frac{27}{8}\) \(w.r.t.\) water, the value of \(r\) is\(......R\)

In a Young's double slit experiment, a laser light of \(560\,nm\) produces an interference pattern with consecutive bright fringes' separation of \(7.2\,mm\). Now another light is used to produce an interference pattern with consecutive bright fringes' separation of \(8.1\,mm\). The wavelength of second light is \(......nm\)

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R .
Assertion A : If oxygen ion \(\left(\mathrm{O}^{-2}\right)\) and Hydrogen ion \(\left(\mathrm{H}^{+}\right)\)enter normal to the magnetic field with equal momentum, then the path of \(\mathrm{O}^{-2}\) ion has a smaller curvature than that of \(\mathrm{H}^{+}\).
Reason R : A proton with same linear momentum as an electron will form a path of smaller radius of curvature on entering a uniform magnetic field perpendicularly.
In the light of the above statements, choose the correct answer from the options given below :

A current through a wire depends on time as \(i =\alpha_{0} t +\beta t ^{2}\) where \(\alpha_{0}=20 A / s\) and \(\beta=8 As ^{-2} .\) Find the charge crossed through a section of the wire in \(15 \,s\) (in \(C\))