Two identical glass rods ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ (refractive index = 1.5) have one convex end of radius of curvature 10 cm. They are placed with the curved surfaces at a distance d as shown in the figure, with their axes (shown by the dashed line) aligned. When a point source of light $\mathrm{P}$ is placed inside rod ${\mathrm{S}}_1$ on its axis at a distance of 50 cm from the curved face, the light rays emanating from it are found to be parallel to the axis inside ${\mathrm{S}}_2$ . The distance $\mathrm{d}$ is

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In a thin rectangular metallic strip a constant current \(I\) flows along the positive \(x\)-direction, as shown in the figure. The length, width and thickness of the strip are \(l, w\) and \(d\), respectively.

A uniform magnetic field \(\vec{B}\) is applied on the strip along the positive \(y\)-direction. Due to this, the charge carriers experience a net deflection along the \(z\)-direction. This results in accumulation of charge carriers on the surface \(P Q R S\) and appearance of equal and opposite charges on the face opposite to \(P Q R S\). A potential difference along the \(z\)-direction is thus developed. Charge accumulation continues until the magnetic force is balanced by the electric force. The current is assumed to be uniformly distributed on the cross section of the strip and carried by electrons.






Question:

Consider two different metallic strips (\(1\) and \(2\)) of the same material. Their lengths are the same, widths are \(w_{1}\) and \(w_{2}\) and thicknesses are \(d_{1}\) and \(d_{2}\), respectively. Two points \(K\) and \(M\) are symmetrically located on the opposite faces parallel to the \(x-y\) plane (see figure). \(V_{1}\) and \(V_{2}\) are the potential differences between \(K\) and \(M\) in strips \(1\) and \(2\), respectively. Then, for a given current \(I\) flowing through them in a given magnetic field strength \(B\), the correct statement(s) is(are)

Let ${\mathrm{E}}_1\left(\mathrm{r}\right),\mathrm{ }{\mathrm{E}}_2\left(\mathrm{r}\right)$ and ${\mathrm{E}}_3\left(\mathrm{r}\right)$ be the respective electric fields at a distance r from a point charge Q, an infinitely long wire with constant linear charge density $\mathrm{\lambda }$ , and an infinite plane with uniform surface charge density $\mathrm{\sigma }$ . If ${\mathrm{E}}_1\left({\mathrm{r}}_0\right)={\mathrm{E}}_2\left({\mathrm{r}}_0\right)={\mathrm{E}}_3\left({\mathrm{r}}_0\right)$ at a given distance ${\mathrm{r}}_0$ , then

A particle, of mass ${10}^{-3}\mathrm{ kg}$ and charge $1.0 \mathrm{C}$, is initially at rest. At time $t=0$, the particle comes under the influence of an electric field $\overrightarrow{E}\left(t\right)=E_0\sin \left(\omega t\right) \hat{\mathrm{i}}$ where $E_0=1.0\mathrm{ N} {\mathrm{C}}^{-1}$ and $\omega ={10}^3\mathrm{ rad} {\mathrm{s}}^{-1}$. Consider the effect of only the electrical force on the particle. Then the maximum speed, in $\mathrm{m} {\mathrm{s}}^{-1}$, attained by the particle at subsequent times is ____________.

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When liquid medicine of density \(\rho\) is to be put in the eye, it is done with the help of a dropper. As the bulb on the top of the dropper is pressed, a drop forms at the opening of the dropper. We wish to estimate the size of the drop.
We first assume that the drop formed at the opening is spherical because that requires a minimum increase in its surface energy. To determine the size, we calculate the net vertical force due to the surface tension \(T\) when the radius of the drop is \(R\). When this force becomes smaller than the weight of the drop, the drop gets detached from the dropper.
Question :
If \(r=5 \times 10^{-4} \mathrm{~m}, \rho=10^3 \mathrm{~kg} \mathrm{~m}^{-3}\), \(g=10 \mathrm{~ms}^{-2}, T=0.11 \mathrm{Nm}^{-1}\), the radius of the drop when it detaches from the dropper is approximately

The figures below depict two situations in which two infinitely long static line charges of constant positive line charge density $\mathrm{\lambda }$ are kept parallel to each other. In their resulting electric field, point charges $\mathrm{q}$ and $-\mathrm{q}$ are kept in equilibrium between them. The point charges are confined to move in the $\mathrm{x}$ direction only. If they are given a small displacement about their equilibrium positions, then the correct statements(s) is (are)

An ideal gas is undergoing a cyclic thermodynamic process in different ways as shown in the corresponding P-V diagrams in column 3 of the table. Consider only the path from state 1 to state 2. W denotes the corresponding work done on the system. The equations and plots in the table have standard notations as used in thermodynamic process. Here $\gamma$ is the ratio of heat capacities at constant pressure and constant volume. The number of moles in the gas is n.

Column – 1	Column – 2	Column – 3
(I) \(W_{1 \rightarrow 2}=\frac{1}{\gamma-1}\) \(\left(P_2 V_2-P_1 V_1\right)\)	(i) Isothermal	(P)
(II) \(W_{1 \rightarrow 2}=\) \(-P V_2+P V_1\)	(ii) Isochoric	(Q)
(III) \(W_{1 \rightarrow 2}=0\)	(iii) Isobaric	(R)
(IV) \(W_{1 \rightarrow 2}=\) \(-n R T \ln \left(\frac{V_2}{V_1}\right)\)	(iv) Adiabatic	(S)

Which of the following options is the only correct representation of the process $∆U=∆Q-P∆V$?

If \(\vec{a}, \vec{b}\) and \(\vec{c}\) are unit vectors satisfying \(|\vec{a}-\vec{b}|^{2}+|\vec{b}-\vec{c}|^{2}+|\vec{c}-\vec{a}|^{2}=9\), then \(|2 \vec{a}+5 \vec{b}+5 \vec{c}|\) is

Consider the following compounds in the liquid form:
\(O_2, HF, H_2 O, NH_3, H_2 O_2, CCl_4, CHCl_3,\) \(C_6 H_6, C_6 H_5 Cl .\)
When a charged comb is brought near their flowing stream, how many of them show deflection as per the following figure?

\[
\text { Match the statements given in Column I with the values given in Column II. }
\]

A nuclear power plant supplying electrical power to a village uses a radioactive material of half life $\mathrm{T}$ years as the fuel. The amount of fuel at the beginning is such that the total power requirement of the village is 12.5% of the electrical power available from the plant at that time. If the plant is able to meet the total power needs of the village for a maximum period of $\mathrm{n}\mathrm{T}$ years, then the value of $\mathrm{n}$ is

$\mathrm{F}{\mathrm{e}}^{3+}$ is reduced to $\mathrm{F}{\mathrm{e}}^{2+}$ by using

The number of stereoisomers obtained by bromination of trans-2-butene is