The ratio for the speed of the electron in the \(3^{ rd }\) orbit of \(He ^{+}\)to the speed of the electron in the \(3^{\text {ad }}\) orbit of hydrogen atom will be

A screw gauge with a pitch of \(0.5 \ mm\) and a circular scale with \(50\) divisions is used to measure the thickness of a thin sheet of Aluminium. Before starting the measurement, it is found that wen the two jaws of the screw gauge are brought in contact, the \(45^{th} \)division coincides with the main scale line and the zero of the main scale is barely visible. What is the thickness of the sheet (in \(mm\)) if the main scale reading is \(0.5\ mm\) and the \(25^{th}\) division coincides with the main scale line 

The magnitude of magnetic induction at mid-point \(O\) due to current arrangement as shown in Fig will be

If Surface tension \((S)\), Moment of Inertia \((I)\) and Planck’s constant \((h)\), were to be taken as the fundamental units, the dimensional formula for linear momentum would be

In the figure, a very large plane sheet of positive charge is shown. \(P _{1}\) and \(P _{2}\) are two points at distance \(l\) and \(2 \,l\) from the charge distribution. If \(\sigma\) is the surface charge density, then the magnitude of electric fields \(E_{1}\) and \(E_{2}\) at \(P _{1}\) and \(P _{2}\) respectively are

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 _______ \(\Omega\).

A \(10\, \Omega, 20 \; mH\) coil carrying constant current is connected to a battery of \(20 \; V\) through a switch is opened current becomes zero in \(100 \; \mu s\). The average emf induced in the coil is \(\dots \; V\).

The angle between vector \((\overrightarrow{{A}})\) and \((\overrightarrow{{A}}-\overrightarrow{{B}})\) is :

The radius of a nucleus of mass number \(64\) is \(4.8\) fermi. Then the mass number of another nucleus having radius of \(4\) fermi is \(\frac{1000}{x}\), where \(x\) is _______.

A hoop of radius \(r\) and mass \(m\) rotating with an angular velocity \(\omega_0\) is placed on a rough horizontal surface. The initial velocity of the centre of the hoop is zero. What will be the velocity of the centre of the hoop when it ceases to slip?

A uniform sphere of mass \(500\; g\) rolls without slipping on a plane horizontal surface with its centre moving at a speed of \(5.00\; \mathrm{cm} / \mathrm{s}\). Its kinetic energy is

\(50\, W/m^2\) energy density of sunlight is normally incident on the surface of a solar panel. Some part of incident energy \((25\%)\) is reflected from the surface and the rest is absorbed. The force exerted on \(1\, m^2\) surface area will be close to \((c = 3 \times10^8\, m / s)\)