Current Affairs JEE Main & Advanced

If two non zero vectors are represented by the two sides of a triangle taken in same order then the resultant is given by the closing side of triangle in opposite order. i.e.\[\overrightarrow{R}=\overrightarrow{A}+\overrightarrow{B}\] Q \[\overrightarrow{OB}=\overrightarrow{OA}+\overrightarrow{AB}\]               (1) Magnitude of resultant vector In \[\Delta \,ABN,\]\[\cos \theta =\frac{AN}{B}\]\[\therefore \]\[AN=B\cos \theta \]            \[\sin \theta =\frac{BN}{B}\]  \[\therefore \] \[BN=B\sin \theta \] In \[\Delta OBN,\] we have \[O{{B}^{2}}=O{{N}^{2}}+B{{N}^{2}}\]   \[\Rightarrow \]\[{{R}^{2}}={{(A+B\cos \theta )}^{2}}+{{(B\sin \theta )}^{2}}\]  \[\Rightarrow \]\[{{R}^{2}}={{A}^{2}}+{{B}^{2}}{{\cos }^{2}}\theta +2AB\cos \theta +{{B}^{2}}{{\sin }^{2}}\theta \] \[\Rightarrow \]\[{{R}^{2}}={{A}^{2}}+{{B}^{2}}({{\cos }^{2}}\theta +{{\sin }^{2}}\theta )+2AB\cos \theta \] \[\Rightarrow \]\[{{R}^{2}}={{A}^{2}}+{{B}^{2}}+2AB\cos \theta \] \[\Rightarrow \] \[R=\sqrt{{{A}^{2}}+{{B}^{2}}+2AB\cos \theta }\] (2) Direction of resultant vectors : If \[\theta \] is angle between \[\overrightarrow{A}\] and \[\overrightarrow{B,}\] then \[\,|\overrightarrow{A}+\overrightarrow{B}|\,=\sqrt{{{A}^{2}}+{{B}^{2}}+2AB\cos \theta }\] If \[\overrightarrow{R}\]makes an angle a with \[\overrightarrow{A},\] then in \[\Delta OBN,\] \[\tan \alpha =\frac{BN}{ON}=\frac{BN}{OA+AN}\] \[\tan \alpha =\frac{B\sin \theta }{A+B\cos \theta }\]      

(1) Schrodinger wave equation is given by Erwin Schrödinger in 1926 and based on dual nature of electron. (2) In it electron is described as a three dimensional  wave in the electric field of a positively charged nucleus. (3) The probability of finding an electron at any point around the nucleus can be determined by the help of Schrodinger wave equation which is, \[\frac{\partial {}^{2}\Psi }{\partial x{}^{2}}+\,\frac{\partial {}^{2}\Psi }{\partial y{}^{2}}+\,\frac{\partial {}^{2}\Psi }{\partial z{}^{2}}+\frac{8\pi {}^{2}m}{h{}^{2}}(E-V)\,\Psi =0\]            Where \[x,\,y\] and z are the 3 space co-ordinates, m = mass of electron, h = Planck?s constant, E = Total energy, V  = potential energy of electron, \[\Psi \]= amplitude of wave also called as wave function, \[\partial \] = for an infinitesimal change.            (4) The Schrodinger wave equation  can also be written as, \[\nabla {}^{2}\Psi +\frac{8\pi {}^{2}m}{h{}^{2}}(E-V)\,\,\Psi =0\]         Where \[\nabla \]= laplacian operator.            (5) Physical significance of \[\Psi \] and \[\Psi {}^{2}\]            (i) The wave function \[\Psi \] represents the amplitude of the electron wave. The amplitude \[\Psi \] is thus a function of space  co-ordinates and time i.e. \[\Psi =\Psi (x,\,y,\,\,z......times)\]            (ii) For a single particle, the square of the wave function \[(\Psi {}^{2})\] at any point is proportional to the probability  of finding the particle at that point.            (iii) If \[\Psi {}^{2}\] is maximum than probability of finding \[{{e}^{-}}\] is maximum around nucleus and the place where probability  of finding \[e{}^{-}\] is maximum is called electron density, electron cloud or an atomic orbital. It is different from the Bohr?s orbit.            (iv) The solution of this equation provides a set of number called quantum numbers which describe specific or definite energy state of the electron in atom and information about the shapes and orientations of the most probable distribution of electrons around the nucleus.          Radial probability distribution curves : Radial probability is \[R=4\pi {{r}^{2}}dr{{\psi }^{2}}.\] The plats of \[R\]  distance from nucleus as follows                

(1) Equal vectors : Two vectors \[\overrightarrow{A}\] and \[\overrightarrow{B}\] are said to be equal when they have equal magnitudes and same direction. (2) Parallel vector : Two vectors \[\overrightarrow{A}\] and \[\overrightarrow{B}\] are said to be parallel when (i) Both have same direction. (ii) One vector is scalar (positive) non-zero multiple of another vector. (3) Anti-parallel vectors : Two  vectors \[\overrightarrow{A}\] and \[\overrightarrow{B}\] are said to be anti-parallel when (i) Both have opposite direction. (ii) One vector is scalar non-zero negative multiple of another vector. (4) Collinear vectors : When the vectors under consideration can share the same support or have a common support then the considered vectors are collinear. (5) Zero vector \[(\overrightarrow{0})\]: A vector having zero magnitude and arbitrary direction (not known to us) is a zero vector. (6) Unit vector : A vector divided by its magnitude is a unit vector. Unit vector for \[\overrightarrow{A}\] is \[\hat{A}\] (read as A cap or A hat). Since, \[\hat{A}=\frac{\overrightarrow{A}}{A}\] \[\Rightarrow \]\[\overrightarrow{A}=A\,\hat{A}\]. Thus, we can say that unit vector gives us the direction. (7) Orthogonal unit vectors  \[\hat{i}\,,\,\hat{j}\] and \[\hat{k}\]are called orthogonal unit vectors. These vectors must form a Right Handed Triad (It is a coordinate system such that when we Curl the fingers of right hand from x to y then we must get the direction of z along thumb). The \[\hat{i}=\frac{\overrightarrow{x}}{x}\],\[\hat{j}=\frac{\overrightarrow{y}}{y}\],\[\hat{k}=\frac{\overrightarrow{z}}{z}\] \ \[\overrightarrow{x}=x\hat{i}\], \[\overrightarrow{y}=y\hat{j}\], \[\overrightarrow{z}=z\hat{k}\]         (8) Polar vectors : These have starting point or point of application . Example displacement and force etc. (9) Axial Vectors : These represent rotational effects and are always along the axis of rotation in accordance with right hand screw rule. Angular velocity, torque and angular momentum, etc., are example of physical quantities of this type.     (10) Coplanar vector : Three (or more) vectors are called coplanar vector if they lie in the same plane. Two (free) vectors are always coplanar.

Physical quantities having magnitude, direction and obeying laws of vector algebra are called vectors. Example: Displacement, velocity, acceleration, momentum, force, impulse, weight, thrust, torque, angular momentum, angular velocity etc. If a physical quantity has magnitude and direction both, then it does not always imply that it is a vector. For it to be a vector the third condition of obeying laws of vector algebra has to be satisfied. Example: The physical quantity current has both magnitude and direction but is still a scalar as it disobeys the laws of vector algebra.                   

This principle  states ?It is impossible to specify at any given moment both the position and momentum (velocity) of an electron?.            Mathematically it is represented as , \[\Delta x\,.\,\Delta p\ge \frac{h}{4\pi }\]            Where \[\Delta x=\]uncertainty is position of the particle, \[\Delta p=\] uncertainty in the momentum of the particle            Now since \[\Delta p=m\,\Delta v\]            So equation becomes,\[\Delta x.\,m\Delta v\ge \frac{h}{4\pi }\] or  \[\Delta x\,\times \,\Delta v\ge \frac{h}{4\pi m}\]            In terms of uncertainty in energy, \[\Delta E\] and uncertainty in time \[\Delta t,\] this principle is written as,   \[\Delta E\,.\,\Delta t\ge \frac{h}{4\pi }\]

(1) In 1924, the French physicist, Louis de Broglie suggested that if light has both particle and wave like nature, the similar duality must be true for matter. Thus an electron, behaves both as a material particle and as a wave. (2) This presented a new wave mechanical theory of matter. According to this theory, small particles like electrons when in motion possess wave properties. (3) According to de-broglie, the wavelength associated with a particle of mass m, moving with velocity v is given by the relation   \[\lambda \,=\,\frac{h}{mv},\]  where h = Planck’s constant. (4) This can be derived as follows according to Planck’s equation,  \[E=\,h\nu =\frac{h.c}{\lambda }\]  \[\left( \because \ \ \nu =\frac{c}{\lambda } \right)\] energy of  photon (on the basis of Einstein’s mass energy relationship),  \[E=\,mc{}^{2}\]  Equating both \[\frac{hc}{\lambda }=\,\,mc{}^{2}\,\,or\,\,\lambda =\frac{h}{mc}\]  which is same as de-Broglie relation.   \[\left( \because \ \ mc=p \right)\] (5) This was experimentally verified by Davisson and Germer by observing diffraction effects with an electron beam.  Let the electron is accelerated with a potential of V than the Kinetic energy is  \[\frac{1}{2}mv{}^{2}=\,\,eV\];  \[m{}^{2}v{}^{2}=\,\,2eVm\]  \[mv=\sqrt{2eVm}=\,\,P\];  \[\lambda =\frac{h}{\sqrt{2eVm}}\] (6) If Bohr’s theory is associated with de-Broglie’s equation then wave length of an electron can be determined in bohr’s orbit and relate it with circumference and multiply with a whole number \[2\pi r=n\lambda \,\,or\,\,\lambda =\frac{2\pi r}{n}\] From de-Broglie equation, \[\lambda =\frac{h}{mv}\].  Thus \[\frac{h}{mv}=\frac{2\pi r}{n}\] or \[mvr=\frac{nh}{2\pi }\] (7) The de-Broglie equation is applicable to all material objects but it has significance only in case of microscopic particles. Since, we come across macroscopic objects in our everyday life, de-broglie relationship has no significance in everyday life.

Bohr retained the essential features of the Rutherford model of the atom. However, in order to account for the stability of the atom he introduced the concept of the stationary orbits. The Bohr postulates are,            (1) An atom consists of positively charged nucleus responsible for almost the entire mass of the atom (This assumption is retention of Rutherford model). (2) The electrons revolve around the nucleus in certain permitted circular orbits of definite radii. (3) The permitted orbits are those for which the angular momentum of an electron is an intergral multiple of \[h/2\pi \] where \[h\] is the Planck?s constant. If \[m\] is the mass and \[v\] is the velocity of the electron in a permitted orbit of radius \[r,\] then \[L=mvr=\frac{nh}{2\Pi }\]; \[n=1\], 2, 3, ??\[\infty \] Where \[L\] is the orbital angular momentum and \[n\] is the number of orbit. The integer \[n\] is called the principal quantum number. This equation is known as the Bohr quantization postulate. (4) When electrons move in permitted discrete orbits they do not radiate or lose energy. Such orbits are called stationary or non-radiating orbits. In this manner, Bohr overcame Rutherford?s difficulty to account for the stability of the atom. Greater the distance of energy level from the nucleus, the more is the energy associated with it.  The different energy levels were numbered as 1,2,3,4 .. and called as \[K,\,L,\,M,\,N,\]?. etc. (5) Ordinarily an electron continues to move in a particular stationary state or orbit. Such a state of atom is called ground state. When energy is given to the electron it jumps to any higher energy level and is said to be in the excited state. When the electron jumps from higher to lower energy state, the energy is radiated. Advantages of Bohr?s theory (i) Bohr?s theory satisfactorily explains the spectra of species having one electron, viz. hydrogen atom, \[H{{e}^{+}},L{{i}^{2+}}\]etc. (ii) Calculation of radius of Bohr?s orbit : According to Bohr, radius of  nth orbit in which electron moves is \[{{r}_{n}}=\left[ \frac{{{h}^{2}}}{4{{\pi }^{2}}m{{e}^{2}}k} \right].\frac{{{n}^{2}}}{Z}\]         Where, \[n=\]Orbit number, \[m=\]Mass number \[\left[ 9.1\times {{10}^{-31}}kg \right]\,,\]\[e=\]Charge on the electron \[\left[ 1.6\times {{10}^{-19}} \right]\] \[Z=\]Atomic number of element, k = Coulombic constant \[\left[ 9\times {{10}^{9}}N{{m}^{2}}{{c}^{-2}} \right]\] After putting the values of m,e,k,h, we get.                     \[{{r}_{n}}=\frac{{{n}^{2}}}{Z}\times 0.529\overset{{}^\circ }{\mathop{\text{A}}}\,\] (iii) Calculation of velocity of electron \[{{V}_{n}}=\frac{2\pi {{e}^{2}}ZK}{nh},\,{{V}_{n}}={{\left[ \frac{Z{{e}^{2}}}{mr} \right]}^{1/2}}\];\[{{V}_{n}}=\frac{2.188\times {{10}^{8}}Z}{n}cm.{{\sec }^{-1}}\] (iv) Calculation of energy of electron in Bohr's orbit Total energy of electron = K.E. + P.E. of electron \[=\frac{kZ{{e}^{2}}}{2r}-\frac{kZ{{e}^{2}}}{r}=-\frac{kZ{{e}^{2}}}{2r}\] Substituting of r, gives us \[E=\frac{-2{{\pi }^{2}}\,m{{Z}^{2}}{{e}^{4}}{{k}^{2}}}{{{n}^{2}}{{h}^{2}}}\] Where, n=1, 2, 3???.\[\infty \] Putting the value of m, e, k, h,\[\pi \]we get \[E=21.8\times {{10}^{-12}}\times \frac{{{Z}^{2}}}{{{n}^{2}}}erg\,per\,atom\]    \[=-21.8\times {{10}^{-19}}\times \frac{{{Z}^{2}}}{{{n}^{2}}}J\,per\,atom\,(1J=\text{1}{{\text{0}}^{\text{7}}}erg)\]                    \[E=-13.6\times \frac{{{Z}^{2}}}{{{n}^{2}}}eV\,per\,atom\text{(1eV}=\text{1}\text{.6}\times \text{1}{{\text{0}}^{-19}}J)\]    \[=-13.6\times \frac{{{Z}^{2}}}{{{n}^{2}}}k.cal/mole\] (1 cal = 4.18J) or \[\frac{-1312}{{{n}^{2}}}{{Z}^{2}}kJmo{{l}^{-1}}\] When an electron jumps from an outer orbit (higher energy) \[{{n}_{2}}\]to an inner orbit (lower energy)\[{{n}_{1}},\]then the energy emitted in form of radiation is given by \[\Delta E={{E}_{{{n}_{2}}}}-{{E}_{{{n}_{1}}}}=\frac{2{{\pi }^{2}}{{k}^{2}}m{{e}^{4}}{{Z}^{2}}}{{{h}^{2}}}\left( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right)\,\] \[\Rightarrow \ \Delta E=13.6{{Z}^{2}}\left( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right)eV/atom\] As we know that \[E=h\bar{\nu },\]\[c=\nu \lambda \]and \[\bar{\nu }=\frac{1}{\lambda }\] \[=\frac{\Delta E}{hc},\] \[=\frac{2{{\pi more...

(1) When radiations with certain minimum frequency \[({{\nu }_{0}})\] strike the surface of a metal, the electrons are ejected from the surface of the metal. This phenomenon is called photoelectric effect and the electrons emitted are called photo-electrons. The current constituted by photoelectrons is known as photoelectric current. (2) The electrons are ejected only if the radiation striking the surface of the metal has at least a minimum frequency \[({{\nu }_{0}})\] called Threshold frequency. The minimum potential at which the plate photoelectric current becomes zero is called stopping potential. (3) The velocity or kinetic energy of the electron ejected depend upon the frequency of the incident radiation and is independent of its intensity. (4) The number of photoelectrons ejected is proportional to the intensity of incident radiation. (5) Einstein?s photoelectric effect equation According to Einstein, Maximum kinetic energy of the ejected electron = absorbed energy ? threshold energy            \[\frac{1}{2}mv_{\max }^{2}=h\nu -h{{\nu }_{0}}=hc\left[ \frac{1}{\lambda }-\frac{1}{{{\lambda }_{0}}} \right]\]                 Where, \[{{\nu }_{0}}\] and \[{{\lambda }_{0}}\] are threshold frequency and threshold wavelength.

When black body is heated, it emits thermal radiation?s of different wavelengths or frequency. To explain these radiations, max planck put forward a theory known as planck?s quantum theory.             (i) The radiant energy which is emitted or absorbed by the black body is not continuous but discontinuous in the form of small discrete packets of energy, each such packet of energy is called a 'quantum'. In case of light, the quantum of energy is called a 'photon'. (ii) The energy of each quantum is directly proportional to the frequency (\[\nu \]) of the radiation, i.e.         \[E\propto \nu \] or \[E=hv=\frac{hc}{\lambda }\] Where, \[h=\] Planck's constant = 6.62×10?27 erg. sec. or \[6.62\times {{10}^{-34}}Joules\,\sec .\] (iii) The total amount of energy emitted or absorbed by a body will be some whole number quanta. Hence \[E=nh\nu ,\] where n is an integer.

(1) Rutherford carried out experiment on the bombardment of thin (10?4 mm) Au foil with high speed positively charged \[\alpha -\]particles emitted from Ra and gave the following observations based on this experiment, (i) Most of the \[\alpha -\] particles passed without any deflection. (ii) Some of them were deflected away from their path. (iii) Only a few (one in about 10,000) were returned back to their original direction of propagation.                                   (2) From the above observations he concluded that, an atom consists of (i) Nucleus which is small in size but carries the entire mass i.e. contains all the neutrons and protons. (ii) Extra nuclear part which contains electrons. This model was similar to the solar system. (3) Properties of the nucleus (i) Nucleus is a small, heavy, positively charged portion of the atom and located at the centre of the atom. (ii) All the positive charge of atom (i.e. protons) are present in nucleus. (iii) Nucleus contains neutrons and protons, and hence these particles collectively are also referred to as nucleons.            (iv) The size of nucleus is measured in Fermi (1 Fermi = 10?13 cm). (v) The radius of nucleus is of the order of \[1.5\times {{10}^{-13}}cm.\] to \[6.5\times {{10}^{-13}}cm.\] i.e. \[1.5\] to \[6.5\] Fermi. Generally the radius of the nucleus (\[{{r}_{n}})\] is given by the following relation,          \[{{r}_{n}}={{r}_{o}}(=1.4\times {{10}^{-13}}cm)\times {{A}^{1/3}}\]           This exhibited that nucleus is \[{{10}^{-5}}\] times small in size as compared to the total size of atom. (vi) The Volume of the nucleus is about \[{{10}^{-39}}\]\[c{{m}^{3}}\] and that of atom is \[{{10}^{-24}}c{{m}^{3}},\] i.e., volume of the nucleus is \[{{10}^{-15}}\] times that of an atom. (vii) The density of the nucleus is of the order of \[{{10}^{15}}g\,c{{m}^{-3}}\] or \[{{10}^{8}}\] tonnes \[c{{m}^{-3}}\] or \[{{10}^{12}}kg/cc\]. If nucleus is spherical than, Density =\[\frac{\text{mass of the nucleus}}{\text{volume of the nucleus  }}=\]\[\frac{\text{mass number}}{6.023\times {{10}^{23}}\times \frac{4}{3}\pi {{r}^{3}}}\] (4) Drawbacks of Rutherford's model (i) It does not obey the Maxwell theory of electrodynamics, according to it ?A small charged particle moving around an oppositely charged centre continuously loses its energy?. If an electron does so, it should also continuously lose its energy and should set up spiral motion ultimately failing into the nucleus. (ii) It could not explain the line spectra of \[H-\] atom and discontinuous spectrum nature.