A charged particle is introduced at the origin $x=0, y=0, z=0$ with a given initial velocity $\overrightarrow{v}$ . A uniform electric field $\overrightarrow{E}$ and a uniform magnetic field $\overrightarrow{B}$ exist everywhere. The velocity $\overrightarrow{v}$ , electric field $\overrightarrow{E}$ and a uniform magnetic field $\overrightarrow{B}$ are given in the columns below

Column 1	Column 2	Column 3
1. Electron with $\overrightarrow{v}=2\frac{E_0}{B_0}\widehat{x}$	(i) $\overrightarrow{E}=E_0\widehat{z}$	(P) $\overrightarrow{B}=-B_0\widehat{x}$
2. Electron with $\overrightarrow{v}=2\frac{E_0}{B_0}\widehat{y}$	(ii) $\overrightarrow{E}=-E_0\widehat{y}$	(Q) $\overrightarrow{B}=B_0\widehat{x}$
3.Proton with $\overrightarrow{v}$ =0	(iii) $\overrightarrow{E}={-E}_0\widehat{x}$	(R) $\overrightarrow{B}=-B_0\widehat{y}$
4. Proton with $\overrightarrow{v}=2\frac{E_0}{B_0}\widehat{x}$	(iv) $\overrightarrow{E}=E_0\widehat{x}$	(S) $\overrightarrow{B}=-B_0\widehat{z}$

In which case will the particle describe a helical path with axis along positive z direction?

A proton is fired from very far away towards a nucleus with charge \(Q=120 e\), where \(e\) is the electronic charge. It makes a closest approach of \(10 \mathrm{fm}\) to the nucleus. The de Broglie wavelength (in units of \(\mathrm{fm}\) ) of the proton at its start is: (take the proton mass, \(m_{p}=(5 / 3) \times 10^{-27} \mathrm{~kg} ; h / e=4.2 \times\) \(10^{-15} \mathrm{~J} . \mathrm{S} / \mathrm{C}\);

\(\frac{1}{4 \pi \varepsilon_{0}}=9 \times 10^{9} \mathrm{~m} / \mathrm{F} ; 1 \mathrm{fm}=10^{-15} \mathrm{~m}\)

A particle of mass M and positive charge Q, moving with a constant velocity ${\overrightarrow{\text{u}}}_1=4\hat{\text{i}}{\text{ms}}^{-1}$, enters a region of uniform static magnetic field normal to the x-y plane. The region of the magnetic field extends from x = 0 to x = L for all values of y. After passing through this region, the particle emerges on the other side after 10 milliseconds with a velocity ${\overrightarrow{\text{u}}}_2=2\left(\sqrt{3}\hat{\text{i}}+\hat{\text{j}}\right){\text{ms}}^{-1}$. The correct statement (s) is (are)

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.

A radioactive sample \(S_1\) having an activity of \(5 \mu \mathrm{Ci}\) has twice the number of nuclei as another sample \(S_2\) which has an activity of \(10 \mu \mathrm{Ci}\). The half lives of \(S_1\) and \(S_2\) can be

Young's double slit experiment is carried out by using green, red and blue light, one color at a time. The fringe widths recorded are \(\beta_{G}, \beta_{R}\) and \(\beta_{B}\), respectively. Then,

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A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let \(N\) be the number density of free electrons, each of mass \(m\). When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions.If the electric field becomes zero, the electrons being to oscillate about the positive ions with a natural angular frequency \(\omega_p\), which is called the plasma frequency. To sustain the oscillations, a time varying electric field needs to be applied that has an angular frequency \(\omega\), where a part of the energy is absorbed and a part of it is reflected. As \(\omega\) approaches \(\omega_p\), all the free electrons are set to resonance together and all the energy is reflected. This is the explanation of high reflectivity of metals.
Question :
Estimate the wavelength at which plasma reflection will occur for a metal having the density of electrons \(N \approx 4 \times 10^{27} \mathrm{~m}^{-1}\). Take \(\varepsilon_0=10^{-11}\) and \(m=10^{-30}\), where these quantities are in proper SI units.

At room temperature, disproportionation of an aqueous solution of in situ generated nitrous acid \(\left(\mathrm{HNO}_2\right)\) gives the species

A particle of mass \(m\) and charge \(q\), moving with velocity \(v\) enters Region II normal to the boundary as shown in the figure. Region II has a uniform magnetic field \(B\) perpendicular to the plane of the paper. The length of the Region II is \(l\). Choose the correct choice(s)

A small electric dipole \(\vec{p}_0\), having a moment of inertia \(I\) about its center, is kept at a distance \(r\) from the center of a spherical shell of radius \(R\). The surface charge density \(\sigma\) is uniformly distributed on the spherical shell. The dipole is initially oriented at a small angle \(\theta\) as shown in the figure. While staying at a distance \(r\), the dipole is free to rotate about its center.

If released from rest, then which of the following statement(s) is(are) correct?
[ \(\varepsilon_0\) is the permittivity of free space.]

Let $ABC$ be the triangle with $AB=1,AC=3$ and $\angle BAC=\frac{\pi }{2}$. If a circle of radius $r>0$ touches the sides $AB,AC$ and also touches internally the circumcircle of the triangle $ABC$, then the value of $r$ is ______. (round-off the value to two decimal place)

The maximum number of isomers (including stereoisomers) that are possible on monochlorination of the following compounds, is

A symmetric star shaped conducting wire loop is carrying a steady state current $I$ as shown in the figure. The distance between the diametrically opposite vertices of the star is $4a$ .