(1) Angular dispersion \[(\theta )\]: Angular separation between extreme colours i.e. \[\theta ={{\delta }_{V}}-{{\delta }_{R}}=\mathbf{(}{{\mu }_{V}}-{{\mu }_{R}}\mathbf{)}A\]. It depends upon \[\mu \] and A.
(2) Dispersive power \[(\omega )\]:
\[\omega =\frac{\theta }{{{\delta }_{y}}}=\frac{{{\mu }_{V}}-{{\mu }_{R}}}{{{\mu }_{y}}-1}\text{ where }\left\{ {{\mu }_{y}}=\frac{{{\mu }_{V}}+{{\mu }_{R}}}{2} \right\}\]
\[\Rightarrow \]It depends only upon the material of the prism i.e. m and it doesn't depends upon angle of prism A
(3) Combination of prisms : Two prisms (made of crown and flint material) are combined to get either dispersion only or deviation only.
(i) Dispersion without deviation (chromatic combination)
\[\frac{A'}{A}=-\frac{({{\mu }_{y}}-1)}{(\mu {{'}_{y}}-1)}\]
\[{{\theta }_{\text{net}}}=\theta \,\left( 1-\frac{\omega '}{\omega } \right)=(\omega \delta -\omega '\delta ')\]
(ii) Deviation without dispersion (Achromatic combination)
\[\frac{A'}{A}=-\frac{({{\mu }_{V}}-{{\mu }_{R}})}{(\mu \,{{'}_{V}}-\mu \,{{'}_{R}})}\]
\[{{\delta }_{\text{net}}}=\delta \,\left( 1-\frac{\omega }{\omega '} \right)\] 
\[A={{r}_{1}}+{{r}_{2}}\] and \[i+e=A+\delta \]
For surface \[AC\,\,\mu =\frac{\sin i}{\sin {{r}_{1}}}\]; For surface AB \[\frac{1}{\mu }=\frac{\sin {{r}_{2}}}{\sin e}\]
(2) Deviation through a prism : For thin prism \[\delta =(\mu -1)A\]. Also deviation is different for different colour light e.g.\[{{\mu }_{R}}<{{\mu }_{V}}\] so \[{{\delta }_{R}}<{{\delta }_{V}}\].
\[{{\mu }_{\text{Flint}}}>{{\mu }_{\text{Crown}}}\] so \[{{\delta }_{F}}>{{\delta }_{C}}\]
(i) Maximum deviation : Condition of maximum deviation is \[\angle \,i={{90}^{o}}\]\[\Rightarrow \]\[{{r}_{1}}=C,\]\[{{r}_{2}}=A-C\] and from Snell's law on emergent surface
\[e={{\sin }^{-1}}\left[ \frac{\sin (A-C)}{\sin \,C} \right]\]
\[{{\delta }_{\max }}=\frac{\pi }{2}+{{\sin }^{-1}}\left[ \frac{\sin (A-C)}{\sin C} \right]-A\]
(ii) Minimum deviation : It is observed if \[\angle i=\angle e\] and \[\angle {{r}_{1}}=\angle {{r}_{2}}=r\], deviation produced is minimum.
(a) Refracted ray inside the prism is parallel to the base of the prism for equilateral and isosceles prisms.
(b) \[r=\frac{A}{2}\] and \[i=\frac{A+{{\delta }_{m}}}{2}\]
(c) \[\mu =\frac{\sin \,i}{\sin \,A/2}\] or \[\mu =\frac{\sin \frac{A+{{\delta }_{m}}}{2}}{\sin A/2}\] (Prism formula).
(3) Condition of no emergence : For no emergence of light, TIR must takes place at the second surface
For TIR at second surface \[{{r}_{2}}>C\]
So \[A>{{r}_{1}}+C\] (From \[A={{r}_{1}}+{{r}_{2}}\])
As maximum value of \[{{r}_{1}}=C\]
So, \[A\ge 2C.\] for any angle of incidence.
If light ray incident normally on first surface i.e. \[\angle i={{0}^{o}}\] it means \[\angle {{r}_{1}}={{0}^{o}}\]. So in this case condition of no emergence from second surface is \[A>C\].
\[\Rightarrow \]\[\sin A>\sin C\]\[\Rightarrow \] \[\sin A>\frac{1}{\mu }\]\[\Rightarrow \] \[\mu >\text{cosec}\,A\] 
\[{{\mu }_{V}}>{{\mu }_{R}}\] so \[{{f}_{R}}>{{f}_{V}}\]
Mathematically chromatic aberration =\[{{f}_{R}}-{{f}_{V}}=\omega {{f}_{y}}\]
\[\omega =\] Dispersive power of lens.
\[{{f}_{y}}=\] Focal length for mean colour \[=\sqrt{{{f}_{R}}{{f}_{V}}}\]
Removal : To remove this defect i.e. for Achromatism we use two or more lenses in contact in place of single lens.
Mathematically condition of Achromatism is : \[\frac{{{\omega }_{1}}}{{{f}_{1}}}+\frac{{{\omega }_{2}}}{{{f}_{2}}}=0\] or \[{{\omega }_{1}}{{f}_{2}}=-{{\omega }_{2}}{{f}_{1}}\]
(2) Spherical aberration : Inability of a lens to form the point image of a point object on the axis is called Spherical aberration.
In this defect all the rays passing through a lens are not focussed at a single point and the image of a point object on the axis is blurred.
Removal : A simple method to reduce spherical aberration is to use a stop before and infront of the lens. (but this method reduces the intensity of the image as most of the light is cut off). Also by using plano-convex lens, using two lenses separated by distance \[d=F-F'\], using crossed lens.
(3) Coma : When the point object is placed away from the principle axis and the image is received on a screen perpendicular to the axis, the shape of the image is like a comet. This defect is called Coma.
It refers to spreading of a point object in a plane \[\bot \] to principle axis.
Removal : It can be reduced by properly designing radii of curvature of the lens surfaces. It can also be reduced by appropriate stops placed at appropriate distances from the lens.
(4) Curvature : For a point object placed off the axis, the image is spread both along and perpendicular to the principal axis. The best image is, in general, obtained not on a plane but on a curved surface. This defect is known as Curvature.
Removal : Astigmatism or the curvature may be reduced by using proper stops placed at proper locations along the axis.
(5) Distortion : When extended objects are imaged, different portions of the object are in general at different distances from the axis. The magnification is not the same for all portions of the extended object. As a result a line object is not imaged into a line but into a curve.
(6) Astigmatism : The spreading of image (of a point object placed away from the principal axis) along the principal axis is called Astigmatism. 
\[{{f}_{m}}=\frac{R}{2},\]\[{{f}_{l}}=\frac{R}{(\mu -1)}\] so \[F=\frac{R}{2\mu }\]
\[{{f}_{m}}=\infty ,\]\[{{f}_{l}}=\frac{R}{(\mu -1)}\] so \[F=\frac{R}{2\,(\mu -1)}\]
(ii) Double convex lens is silvered
Since \[{{f}_{l}}=\frac{R}{2\,(\mu -1)},\,{{f}_{m}}=\frac{R}{2}\] so \[F=\frac{R}{2\,(2\mu -1)}\] 
\[\frac{1}{F}=\frac{1}{{{f}_{1}}}+\frac{1}{{{f}_{2}}}-\frac{d}{{{f}_{1}}{{f}_{2}}}\] and \[P={{P}_{1}}+{{P}_{2}}-d{{P}_{1}}{{P}_{2}}\]
(5) Combination of parts of a lens :
(1) Coefficient of self-induction : Number of flux linkages with the coil is proportional to the current i. i.e. \[N\varphi \propto i\] or \[N\varphi =Li\] (N is the number of turns in coil and \[N\phi -\]total flux linkage). Hence \[L=\frac{N\varphi }{i}\]= coefficient of self-induction.
(2) If \[i=1\,amp,\,\,N=1\] then, \[L=\phi \] i.e. the coefficient of self induction of a coil is equal to the flux linked with the coil when the current in it is 1 amp.
(3) By Faraday's second law induced emf \[e=-N\frac{d\varphi }{dt}\]. Which gives \[e=-L\frac{di}{dt}\] ; If \[\frac{di}{dt}=1\,amp/sec\]then \[|e|\,=L\]
Hence coefficient of self induction is equal to the emf induced in the coil when the rate of change of current in the coil is unity.
(4) Units and dimensional formula of 'L' : It's S.I. unit
\[\frac{weber}{Amp}=\frac{Tesla\times {{m}^{2}}}{Amp}=\frac{N\times m}{Am{{p}^{2}}}=\frac{Joule}{Am{{p}^{2}}}=\frac{Coulomb\times volt}{Am{{p}^{2}}}\]
\[=\frac{volt\times \sec }{amp}=ohm\times \sec \]. But practical unit is henry (H). It's dimensional formula \[[L]=[M{{L}^{2}}{{T}^{-2}}{{A}^{-2}}]\]
(5) Dependence of self inductance (L) : 'L' does not depend upon current flowing or change in current flowing but it depends upon number of turns (N), Area of cross section (A) and permeability of medium \[(\mu )\].
'L' does not play any role till there is a constant current flowing in the circuit. 'L' comes in to the picture only when there is a change in current.
(6) Magnetic potential energy of inductor : In building a steady current in the circuit, the source emf has to do work against of self inductance of coil and whatever energy consumed for this work stored in magnetic field of coil this energy called as magnetic potential energy (U) of coil
\[U=\int_{\,0}^{\,i}{\,Lidi}=\frac{1}{2}L{{i}^{2}}\]; Also \[U=\frac{1}{2}(Li)i=\frac{N\varphi i}{2}\]
(7) The various formulae for L
| Condition | Figure |
| Circular coil \[L=\frac{{{\mu }_{0}}\pi {{N}^{2}}r}{2}\] |  |
| Solenoid \[L=\frac{{{\mu }_{0}}{{\mu }_{r}}{{N}^{2}}A}{l}=\frac{\mu {{N}^{2}}A}{l}(\mu ={{\mu }_{0}}{{\mu }_{r}})\] | more...
(1) A symmetric lens is cut along optical axis in two equal parts. Intensity of image formed by each part will be same as that of complete lens. Focal length is double the original for each part.
(2) A symmetric lens is cut along principle axis in two equal parts. Intensity of image formed by each part will be less compared as that of complete lens. (aperture of each part is \[\frac{1}{\sqrt{2}}\] times that of complete lens). Focal length remains same for each part.
By this method focal length of convex lens is determined.
Consider an object and a screen placed at a distance \[D(>4f)\] apart. Let a lens of focal length f be placed between the object and the screen.
(1) For two different positions of lens two images \[({{I}_{1}}\text{ and }{{I}_{2}})\]of an object are formed at the screen.
(2) Focal length of the lens \[f=\frac{{{D}^{2}}-{{x}^{2}}}{4D}=\frac{x}{{{m}_{1}}-{{m}_{2}}}\]
where \[{{m}_{1}}=\frac{{{I}_{1}}}{O}\,\]; \[{{m}_{2}}=\frac{{{I}_{2}}}{O}\] and \[{{m}_{1}}{{m}_{2}}=1.\]
(3) Size of object \[O=\sqrt{{{I}_{1}}.\,{{I}_{2}}}\]
If a lens (made of glass) of refractive index \[{{\mu }_{g}}\] is immersed in a liquid of refractive index \[{{\mu }_{l}}\], then its focal length in liquid, \[{{f}_{l}}\] is given by \[\frac{1}{{{f}_{l}}}=({{\,}_{l}}{{\mu }_{g}}-1)\,\left( \frac{1}{{{R}_{1}}}-\frac{1}{{{R}_{2}}} \right)\] ......(i)
If \[{{f}_{a}}\] is the focal length of lens in air, then
\[\frac{1}{{{f}_{a}}}=({{\,}_{a}}{{\mu }_{g}}-1)\,\left( \frac{1}{{{R}_{1}}}-\frac{1}{{{R}_{2}}} \right)\] ......(ii)
\[\Rightarrow \]\[\frac{{{f}_{l}}}{{{f}_{a}}}=\frac{{{(}_{a}}{{\mu }_{g}}-1)}{{{(}_{l}}{{\mu }_{g}}-1)}\] (1) If \[{{\mu }_{g}}>{{\mu }_{l}},\] then \[{{f}_{l}}\] and \[{{f}_{a}}\] are of same sign and \[{{f}_{l}}>{{f}_{a}}\].
That is the nature of lens remains unchanged, but it's focal length increases and hence power of lens decreases.
(2) If \[{{\mu }_{g}}={{\mu }_{l}},\] then \[{{f}_{l}}=\infty \]. It means lens behaves as a plane glass plate and becomes invisible in the medium.
(3) If \[{{\mu }_{g}}<{{\mu }_{l}},\] then \[{{f}_{l}}\] and \[{{f}_{a}}\] have opposite signs and the nature of lens changes i.e. a convex lens diverges the light rays and concave lens converges the light rays.
(1) Inductance is that property of electrical circuits which opposes any change in the current in the circuit.
(2) Inductance is inherent property of electrical circuits. It will always be found in an electrical circuit whether we want it or not.
(3) A straight wire carrying current with no iron part in the circuit will have lesser value of inductance.
(4) Inductance is analogous to inertia in mechanics, because inductance of an electrical circuit opposes any change of current in the circuit.
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