Two point charges \(-Q\) and \(+Q / \sqrt{3}\) are placed in the \(x y\)-plane at the origin \((0,0)\) and a point \((2,0)\), respectively, as shown in the figure. This results in an equipotential circle of radius \(R\) and potential \(V=0\) in the \(x y\)-plane with its center at \((b, 0)\). All lengths are measured in meters.

The value of $b$ is ________ meter.

Paragraph:

Light guidance in an optical fiber can be understood by considering a structure comprising of thin solid glass cylinder of refractive index \(n_{1}\) surrounded by a medium of lower refractive index \(n_{2}\). The light guidance in the structure takes place due to successive total internal reflections at the interface of the media \(n_{1}\) and \(n_{2}\) as shown in the figure. All rays with the angle of incidence \(i\) less than a particular value \(i_{m}\) are confined in the medium of refractive index \(n_{1}\). The numerical aperture (NA) of the structure is defined as \(\sin i_{m}\).



Question:

For two structures namely \(S_{1}\) with \(n_{1}=\sqrt{45} / 4\) and \(n_{2}=3 / 2\), and \(S_{2}\) with \(n_{1}=8 / 5\) and \(n_{2}=7 / 5\) and taking the refractive index of water to be \(4 / 3\) and that of air to be \(1\) , the correct option(s) is(are)

An infinitely long thin wire, having a uniform charge density per unit length of \(5 \mathrm{nC} / \mathrm{m}\), is passing through a spherical shell of radius \(1 \mathrm{~m}\), as shown in the figure. A \(10 \mathrm{nC}\) charge is distributed uniformly over the spherical shell. If the configuration of the charges remains static, the magnitude of the potential difference between points \(\mathrm{P}\) and \(\mathrm{R}\), in Volt, is _______ .
[Given: In SI units \(\frac{1}{4 \pi \epsilon_0}=9 \times 10^9, \ln 2=0.7\). Ignore the area pierced by the wire.]

Two metallic rings \(A\) and \(B\), identical in shape and size but having different resistivities \(\rho_A\) and \(\rho_B\), are kept on top of two identical solenoids as shown in the figure. When current \(I\) is switched on in both the solenoids in identical manner, the rings \(A\) and \(B\) jump to heights \(h_A\) and \(h_B\), respectively, with \(h_A>h_B\). The possible relation(s) between their resistivities and their masses \(m_A\) and \(m_B\) is(are)

A thin uniform cylindrical shell, closed at both ends, is partially filled with water. It is floating vertically in water in half-submerged state. If \(\rho_{c}\) is the relative density of the material of the shell with respect to water, then the correct statement is that the shell is

A particle is moved along a path $\mathrm{A}\mathrm{B}-\mathrm{B}\mathrm{C}-\mathrm{C}\mathrm{D}-\mathrm{D}\mathrm{E}-\mathrm{E}\mathrm{F}-\mathrm{F}\mathrm{A},$ as shown in figure, in presence of a force $\overrightarrow{\mathrm{F}}=\left(\mathrm{\alpha }\mathrm{y}\widehat{\mathrm{i}}+2\mathrm{\alpha }\mathrm{x}\widehat{\mathrm{j}}\right)\mathrm{N},$ where $x$ and $\mathrm{y}$ are in meter and $\mathrm{\alpha }=-1\mathrm{N}/{\mathrm{m}}^{-1}.$ The work done on the particle by this force $\overrightarrow{\mathrm{F}}$ will be ____ Joule.

A stationary tuning fork is in resonance with an air column in a pipe. If the tuning fork is moved with a speed of $2 \mathrm{m} {\mathrm{s}}^{-1}$ in front of the open end of the pipe and parallel to it, the length of the pipe should be changed for the resonance to occur with the moving tuning fork. If the speed of sound in air is $320 \mathrm{m} {\mathrm{s}}^{-1},$ the smallest value of the percentage change required in the length of the pipe is ________.

A horizontal stretched string, fixed at two ends, is vibrating in its fifth harmonic according to the equation, \(y ( x , t )=(0.01 m) \sin \left[\left(62.8 m^{-1}\right) x \right]\) \(\cos \left[\left(628 s^{-1}\right) t \right]\). Assuming \(\pi=3.14\), the correct statement(s) is (are)

In an insulated vessel, \(0.05 \mathrm{~kg}\) steam at \(373 \mathrm{~K}\) and \(0.45 \mathrm{~kg}\) of ice at \(253 \mathrm{~K}\) are mixed. Find the final temperature of the mixture (in kelvin).
\( \text { Given, }L_{\text {fusion }} =80 \mathrm{cal} / \mathrm{g}=336 \mathrm{~J} / \mathrm{g},\) \( L_{\text {vaporization }}=540 \mathrm{cal} / \mathrm{g}=2268 \mathrm{~J} / \mathrm{g}, \)
\( S_{\text {ice }} =2100 \mathrm{~J} / \mathrm{kg}, K=0.5 \mathrm{cal} / \mathrm{gK} \text { and } S_{\text {water }}=\) \(4200 \mathrm{~J} / \mathrm{kg}, K=1 \mathrm{cal} / \mathrm{gK} .\)

An infinitely long solid cylinder of radius \(R\) has a uniform volume charge density \(\rho\). It has a spherical cavity of radius \(R / 2\) with its centre on the axis of the cylinder, as shown in the figure. The magnitude of the electric field at the point \(P\), which is at a distance \(2 R\) from the axis of the cylinder, is given by the expression \(\frac{23 \rho R}{16 K \varepsilon_{0}}\). The value of \(k\) is

A solid sphere of radius \(R\) has moment of inertia \(J\) about its qeometrical axis. If it is melted into a disc of radius \(r\) and thickness \(t\). If its moment of inertia about the tangential axis (which is perpendicular to plane of the disc), is also equal to \(I\), then the value of \(r\) is equal to

Let $\mathrm{g}\mathrm{ }:\mathbb{R}\to \mathbb{R}$ be a differentiable function with $\mathrm{g}\left(0\right)=0,\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }{\mathrm{g}}^{\mathrm{\prime }}\left(0\right)=0$ and ${\mathrm{g}}^{\mathrm{\prime }}\left(1\right)\mathrm{ }\ne 0.\mathrm{ }$ Let $\mathrm{f}\left(\mathrm{x}\right)=\mathrm{ }\left\{\begin{array}{cc}\frac{\mathrm{x}}{\left|\mathrm{x}\right|}\mathrm{ }\mathrm{g}\left(\mathrm{x}\right),& \mathrm{x}\mathrm{ }\ne 0\\ 0,& \mathrm{x}=0\end{array}\right.$ and $\mathrm{h}\left(\mathrm{x}\right)={\mathrm{e}}^{\left|\mathrm{x}\right|}$ for all $\mathrm{x}\in \mathbb{R}.$ Let $\left(\mathrm{foh}\right)\mathrm{ }\left(\mathrm{x}\right)$ denote $\mathrm{f}\left(\mathrm{h}\left(\mathrm{x}\right)\right)\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\left(\mathrm{hof}\right)\mathrm{ }\left(\mathrm{x}\right)\mathrm{ }$denote $\mathrm{h}\left(\mathrm{f}\left(\mathrm{x}\right)\right)$ .Then which of the following is (are) true?

Based on VSEPR theory, the number of 90 degree \(\mathrm{F}-\mathrm{Br}-\mathrm{F}\) angles in \(\mathrm{BrF}_5\) is