Current Affairs JEE Main & Advanced

As per the most established theory it is due to the rotation of the earth where by the various charged ions present in the molten state in the core of the earth rotate and constitute a current. (1) The magnetic field of earth is similar to one which would be obtained if a huge magnet is assumed to be buried deep inside the earth at it's centre. (2) The axis of rotation of earth is called geographic axis and the points where it cuts the surface of earth are called geographical poles \[({{N}_{g}},\,Sq)\]. The circle on the earth's surface perpendicular to the geographical axis is called equator. (3) A vertical plane passing through the geographical axis is called geographical meridian. (4) The axis of the huge magnet assumed to be lying inside the earth is called magnetic axis of the earth. The points where the magnetic axis cuts the surface of earth are called magnetic poles. The circle on the earth's surface perpendicular to the magnetic axis is called magnetic equator. (5) Magnetic axis and Geographical axis don't coincide but they make an angle of \[{{17.5}^{o}}\] with each other. (6) Magnetic equator divides the earth into two hemispheres. The hemisphere containing south polarity of earth's magnetism is called northern hemisphere while the other, the southern hemisphere. (7) The magnetic field of earth is not constant but changes irregularly from place to place on the surface of the earth and even at a given place it varies with time too. (8) Direction of earth's magnetic field is from S  (geographical south) to N (geographical north).

(1) Coulombs law in magnetism : The force between two magnetic poles of strength m1 and m2 lying at a distance \[r\] is given by \[F=k.\frac{{{m}_{1}}{{m}_{2}}}{{{r}^{2}}}\]. In S.I. units \[k=\frac{{{\mu }_{0}}}{4\pi }={{10}^{-7}}wb/Amp\times m\], In CGS units \[k=1\] (2) Magnetic field (i) Magnetic field due to an imaginary magnetic pole (Pole strength m) : Is given by \[B=\frac{F}{{{m}_{0}}}\] also\[B=\frac{{{\mu }_{0}}}{4\pi }.\frac{m}{{{d}^{2}}}\] (ii) Magnetic field due to a bar magnet : At a distance r from the centre of magnet (a) On axial position \[{{B}_{a}}=\frac{{{\mu }_{0}}}{4\pi }\frac{2Mr}{{{({{r}^{2}}-{{l}^{2}})}^{2}}}\]; If \[l<<r\] then \[{{B}_{a}}=\frac{{{\mu }_{0}}}{4\pi }\frac{2M}{{{r}^{3}}}\] (b) On equatorial position : \[{{B}_{e}}=\frac{{{\mu }_{0}}}{4\pi }\frac{M}{{{({{r}^{2}}+{{l}^{2}})}^{3/2}}}\]; If \[l<<r\,;\] then \[{{B}_{e}}=\frac{{{\mu }_{0}}}{4\pi }\frac{M}{{{r}^{3}}}\] (c) General position :  In general position for a short bar magnet \[{{B}_{g}}=\frac{{{\mu }_{0}}}{4\pi }\frac{M}{{{r}^{3}}}\sqrt{(3{{\cos }^{2}}\theta +1)}\] (3) Bar magnet in magnetic field : When a bar magnet is left free in an uniform magnetic field, if align it self in the directional field. (i) Torque : \[\tau =MB\sin \theta \Rightarrow \overrightarrow{\tau }=\overrightarrow{M}\times \overrightarrow{B}\] (ii) Work :  \[W=MB(1-\cos \theta )\] (iii) Potential energy : \[U=-MB\cos \theta =-\overrightarrow{M}\,.\,\overrightarrow{B}\]; (\[\theta =\] Angle made by the dipole with the field) (4) Gauss's law in magnetism : Net magnetic flux through any closed surface is always zero i.e. \[\oint{\overrightarrow{B}.\overrightarrow{ds}=0}\]

(1) Magnetic field and magnetic lines of force : Space around a magnetic pole or magnet or current carrying wire within which it's effect can be experienced is defined as magnetic field. Magnetic field can be represented with the help of a set of lines or curves called magnetic lines of force. (2) Magnetic flux \[(\phi )\] and flux density (B) (i) The number of magnetic lines of force passing normally through a surface is defined as magnetic flux \[(\phi )\]. It's S.I. unit is weber (wb) and CGS unit is Maxwell. Remeber \[1\,wb={{10}^{8}}\] Maxwell. (ii) When a piece of a magnetic substance is placed in an external magnetic field the substance becomes magnetised. The number of magnetic lines of induction inside a magnetised substance crossing unit area normal to their direction is called magnetic induction or magnetic flux density \[(\overrightarrow{B}).\] It is a vector quantity. It's SI unit is Tesla which is equal to  \[\frac{wb}{{{m}^{2}}}=\frac{N}{amp\times m}=\frac{J}{amp\times {{m}^{2}}}=\frac{volt\times \sec }{{{m}^{2}}}\] and CGS unit is Gauss. Remember 1 Tesla \[={{10}^{4}}\] Gauss. (3) Magnetic permeability : It is the degree or extent to which magnetic lines of force can enter a substance and is denoted by \[\mu \]. Or characteristic of a medium which allows magnetic flux to pass through it is called it's permeability. e.g. permeability of soft iron is 1000 times greater than that of air. Also \[\mu ={{\mu }_{0}}\,{{\mu }_{r}};\,\] where \[{{\mu }_{0}}=\] absolute permeability of air or free space = \[4\pi \times {{10}^{-7}}tesla\times m/amp.\] and \[{{\mu }_{r}}=\] Relative permeability of the medium = \[\frac{B}{{{B}_{0}}}=\frac{\text{flux}\,\text{density in material}}{\text{flux density in vacuum}}.\] (4) Intensity of magnetising field \[\mathbf{(}\overrightarrow{H}\mathbf{)}\] (magnetising field) : It is the degree or extent to which a magnetic field can magnetise a substance. Also \[H=\frac{B}{\mu }\]. It's SI unit is  \[A/m.=\frac{N}{{{m}^{2}}\times Tesla}=\frac{N}{wb}=\frac{J}{{{m}^{3}}\times Tesla}=\frac{J}{m\times wb}\] It's CGS unit is Oersted. Also 1 Oersted = 80 A/m (5) Intensity of magnetisation (I) : It is the degree to which a substance is magnetised when placed in a magnetic field. It can also be defined as the pole strength per unit cross sectional area of the substance or the induced dipole moment per unit volume. Hence \[l=\frac{m}{A}=\frac{M}{V}.\] It is a vector quantity, it's S.I. unit is Amp/m. (6) Magnetic susceptibility \[({{\chi }_{m}})\] : It is the property of the substance which shows how easily a substance  can be magnetised. It can also be defined as the ratio of intensity of magnetisation (I) in a substance to the magnetic intensity (H) applied to the substance, i.e. \[{{\chi }_{m}}=\frac{I}{H}\]. It is a scalar quantity with no units and dimensions. (7) Relation between permeability and susceptibility : Total magnetic flux density B in a material is the sum of magnetic flux density in vacuum  \[{{B}_{0}}\] produced by magnetising force and magnetic flux density due to magnetisation of material \[{{B}_{m}}\]. i.e. \[B={{B}_{0}}+{{B}_{m}}\]\[\Rightarrow \]\[B={{\mu }_{0}}H+{{\mu }_{0}}I={{\mu }_{0}}(H+I)={{\mu }_{0}}H(1+\]\[{{\chi }_{m}})\]. Also \[{{\mu }_{r}}=(1+{{\chi }_{m}})\]

A bar magnet consist of two equal and opposite magnetic pole separated by a small distance. Poles are not exactly at the ends. The shortest distance between two poles is called effective length \[({{L}_{e}})\] and is less than its geometric length \[({{L}_{g}})\]. For bar magnet \[{{L}_{e}}=2l\] and     \[{{L}_{e}}=(5/6)\,{{L}_{g}}\].  For semi circular magnet \[{{L}_{g}}=\pi R\] and \[{{L}_{e}}=2R\] (1) Directive properties : When a magnet suspended freely it stays in the earth's N-S direction (in magnetic meridian). (2) Monopole concept : If a magnet is Broken into number of pieces, each piece becomes a magnet. This in turn implies that monopoles do not exist. (i.e., ultimate individual unit of magnetism in any magnet is called dipole). (3) For two rods as shown, if both the rods attract in figure (A) and doesn't attract in figure (B) then, Q is a magnetic and P is simple iron rod. Repulsion is sure test of magnetism. (4) Pole strength (m) : The strength of a magnetic pole to attract magnetic materials towards itself is known as pole strength. (i) It is a scalar quantity. (ii) Pole strength of N and S pole of a magnet is conventionally represented by +m and \[-m\] respectively. (iii) It's SI unit is amp \[\times \] m or N/Tesla and dimensions are [LA]. (iv) Pole strength of the magnet depends on the nature of material of magnet and area of cross section. It doesn't depends upon length. (5) Magnetic moment or magnetic dipole moment \[(\overrightarrow{M})\] : It represents the strength of magnet. Mathematically it is defined as the product of the strength of either pole and effective length. i.e. \[\overrightarrow{M}=m(2\overrightarrow{\,l\,})\] (i) It is a vector quantity directed from south to north. (ii) It's S.I. unit \[amp\times {{m}^{2}}\] or N-m / Tesla and dimensions \[[A{{L}^{2}}]\] (6) Cutting of a rectangular bar magnet : Suppose we have a rectangular bar magnet having length, breadth and mass are L, b and w respectively if it is cut in n equal parts along the length as well as perpendicular to the length simultaneously as shown in the figure then Length of each part \[L'=\frac{L}{\sqrt{n}}\], breadth of each part \[b'=\frac{b}{\sqrt{n}}\] , Mass of each part \[w'=\frac{w}{n}\], pole strength of each part \[m'=\frac{m}{\sqrt{n}}\], Magnetic moment of each part \[M'=m'L'=\frac{m}{\sqrt{n}}\times \frac{L}{\sqrt{n}}=\frac{M}{n}\] If initially moment of inertia of bar magnet about the axes passing from centre and perpendicular to it's length is \[I=w\,\left( \frac{{{L}^{2}}+{{b}^{2}}}{12} \right)\] then moment of inertia of each part \[I'=\frac{I}{{{n}^{2}}}\] (7) Cutting of a thin bar magnet : For thin magnet \[b=0\] so \[L'=\frac{L}{n}\], \[w'=\frac{w}{n}\], \[m'=\frac{m}{n}\], \[I'=\frac{I}{{{n}^{3}}}\]

The molecular theory of magnetism was given by Weber and modified later by Ewing. According to this theory. Every molecule of a substance is a complete magnet in itself. However, in an magnetic substance the molecular magnets are randomly oriented to give net zero magnetic moment. On magnetising, the molecular magnets are realigned in a specific direction leading to a net magnetic moment.

In case of current carrying conductor in a magnetic field force experienced by its small length element is \[d\overrightarrow{F}=id\overrightarrow{l\,}\times \overrightarrow{B}\];  \[id\overrightarrow{l\,}\]= current element \[d\overrightarrow{F}=i(d\overrightarrow{l\,}\times \overrightarrow{B})\] Total magnetic force \[\overrightarrow{F}=\int{d\overrightarrow{F}}=\int{i(d\overrightarrow{l\,}\times \overrightarrow{B})}\]. If magnetic field is uniform i.e., \[\overrightarrow{B}=\] constant \[\overrightarrow{F}=i\,[\int{\overrightarrow{dl}}]\times \overrightarrow{B}=i(\overrightarrow{L}\times \overrightarrow{B})\] \[\int_{{}}^{{}}{\overrightarrow{dl}}=\overrightarrow{L}'=\] vector sum of all the length elements from initial to final point. Which is in accordance with the law of vector addition is equal to length vector \[\overrightarrow{L}\] joining initial to final point. (For a straight conductor \[F=Bil\sin \theta \]) Direction of force : The direction of force is always perpendicular to the plane containing \[i\overrightarrow{dl}\] and \[\overrightarrow{B}\] and is same as that of cross-product of two vectors \[(\overrightarrow{A}\times \overrightarrow{B})\] with \[\overrightarrow{A}=i\,\overrightarrow{dl}\]. The direction of force when current element \[i\,\overrightarrow{dl}\] and \[\vec{B}\] are perpendicular to each other can also be determined by applying either of the following rules Fleming's left-hand rule : Stretch the fore-finger, central finger and thumb of left hand mutually perpendicular. Then if the fore-finger points in the direction of field \[\overrightarrow{B}\] and the central in the direction of current i, the thumb will point in the direction of force. Right-hand palm rule : Stretch the fingers and thumb of right hand at right angles to each other. Then if the fingers point in the direction of field \[\overrightarrow{B}\] and thumb in the direction of current i, then normal to the palm will point in the direction of force            

The Phenomenon of producing a transverse emf in a current carrying conductor on applying a magnetic field perpendicular to the direction of the current is called Hall effect. Hall effect helps us to know the nature and number of charge carriers in a conductor. Consider a conductor having electrons as current carriers. The electrons move with drift velocity \[\overrightarrow{v\,}\] opposite to the direction of flow of current Force acting on electron \[{{F}_{m}}=-e(\overrightarrow{v\,}\times \overrightarrow{B}).\] This force acts along x-axis and hence electrons will move towards face (2) and it becomes negatively charged.

Cyclotron is a device used to accelerated positively charged particles (like, a-particles, deutrons etc.) to acquire enough energy to carry out nuclear disintegration etc. It is based on the fact that the electric field accelerates a charged particle and the magnetic field keeps it revolving in circular orbits of constant frequency. It consists of two hollow D-shaped metallic chambers \[{{D}_{1}}\] and \[{{D}_{2}}\] called dees. The two dees are placed horizontally with a small gap separating them. The dees are connected to the source of high frequency electric field. The dees are enclosed in a metal box containing a gas at a low pressure of the order of \[{{10}^{-3}}\] mm mercury. The whole apparatus is placed between the two poles of a strong electromagnet NS as shown in fig. The magnetic field acts perpendicular to the plane of the dees. (1) Cyclotron frequency : Time taken by ion to describe a semicircular path is given by \[t=\frac{\pi \,r}{v}=\frac{\pi \,m}{qB}\] If \[T=\] time period of oscillating electric field then \[T=2t=\frac{2\pi \,m}{qB}\] the cyclotron frequency \[\nu =\frac{1}{T}=\frac{Bq}{2\pi m}\] (2) Maximum energy of particle : Maximum energy gained by the charged particle \[{{E}_{\max }}=\left( \frac{{{q}^{2}}{{B}^{2}}}{2m} \right)\,{{r}^{2}}\] where \[{{r}_{0}}=\] maximum radius of the circular path followed by the positive ion.

When the moving charged particle is subjected simultaneously to both electric field \[\overrightarrow{E}\] and magnetic field \[\overrightarrow{B}\], the moving charged particle will experience electric force \[\overrightarrow{{{F}_{e}}}=q\overrightarrow{E}\] and magnetic force \[\overrightarrow{{{F}_{m}}}=q(\overrightarrow{v}\times \overrightarrow{B})\]; so the net force on it will be \[\overrightarrow{F}=q\mathbf{[}\overrightarrow{E}+\mathbf{(}\overrightarrow{v\,}\times \overrightarrow{B}\mathbf{)]}\]. Which is the famous 'Lorentz-force equation'. Depending on the directions of \[\overrightarrow{v},\,E\] and \[\overrightarrow{B}\] following situations are possible (i) When \[\overrightarrow{v},\,\overrightarrow{E}\] and \[\overrightarrow{B}\] all the three are collinear : In this situation the magnetic force on it will be zero and only electric force will act and so \[\vec{a}=\frac{{\vec{F}}}{m}=\frac{q\vec{E}}{m}\] (ii) The particle will pass through the field following a straight-line path (parallel field) with change in its speed. So in this situation speed, velocity, momentum and kinetic energy all will change without change in direction of motion as shown (iii) \[\overrightarrow{v\,},\,\overrightarrow{E}\] and \[\overrightarrow{B}\] are mutually perpendicular : In this situation if \[\overrightarrow{E}\] and \[\overrightarrow{B}\] are such that \[\overrightarrow{F}=\overrightarrow{{{F}_{e}}}+\overrightarrow{{{F}_{m}}}=0\] i.e., \[\overrightarrow{a}=(\overrightarrow{F}/m)=0\] as shown in figure, the particle will pass through the field with same velocity, without any deviation in path. And in this situation, as \[{{F}_{e}}={{F}_{m}}\] i.e., \[qE=qvB\] \[v=E/B\] This principle is used in 'velocity-selector' to get a charged beam having a specific velocity.  

(1) Straight line : If the direction of a \[\overrightarrow{v}\]  is parallel or antiparallel to \[\overrightarrow{B},\] \[\theta =0\] or \[\theta ={{180}^{o}}\] and therefore \[F=0\]. Hence the trajectory of the particle is a straight line. (2) Circular path : If \[\overrightarrow{v}\] is perpendicular to \[\overrightarrow{B}\] i.e. \[\theta ={{90}^{o}},\] hence particle will experience a maximum magnetic force \[{{F}_{\max }}=qvB\] which act's in a direction perpendicular to the motion of charged particle. Therefore the trajectory of the particle is a circle. (i) In this case path of charged particle is circular and magnetic force provides the necessary centripetal force i.e. \[qvB=\frac{m{{v}^{2}}}{r}\] \[\Rightarrow \] radius of path  \[r=\frac{mv}{qB}=\frac{p}{qB}=\frac{\sqrt{2mK}}{qB}=\frac{\mathbf{1}}{B}\sqrt{\frac{\mathbf{2}mV}{q}}\] where \[p=\] momentum of charged particle and \[K=\] kinetic energy of charged particle (gained by charged particle after accelerating through potential difference V) then \[p=mv=\sqrt{2mK}=\sqrt{2mqV}\] (ii) If T is the time period of the particle then \[T=\frac{2\pi m}{qB}\] (i.e., time period (or frequency) is independent of speed of particle). (3) Helical path : When the charged particle is moving at an angle to the field (other than \[{{0}^{o}},\,\,{{90}^{o}},\] or \[{{180}^{o}}\]). Particle describes a path called helix. (i) The radius of this helical path is  \[\mathbf{r=}\frac{\mathbf{m(vsin\theta )}}{\mathbf{qB}}\] (ii) Time period and frequency do not depend on velocity and so they are given by \[T=\frac{2\pi \,m}{qB}\] and \[\nu =\frac{qB}{2\pi \,m}\] (iii) The pitch of the helix, (i.e., linear distance travelled in one rotation) will be given by \[p=T(v\cos \theta )=2\pi \frac{m}{qB}(v\cos \theta )\] (iv) If pitch value is p, then number of pitches obtained in length l given as Number of pitches\[=\frac{l}{p}\] and time required \[t=\frac{l}{v\cos \theta }\]