Wave Optics

Introduction : 

In 1637;Descattes gave cospuscular theory of light and derived Snell's law . The cospuscular theory predicted that if a ray of light bends towards the normal the speed of light in the second medium would be greater than the first medium .Corpuscular theory was further developed by Newton . In 1678, the Dutch physicist Christian Huygens put forward the wave theory of light .It predicted that if light bends towards the normal its speed in second medium would be less than the first , a sharp contrast to the Corpuscular theory .In 1850, Foucault experimentally proved that speed of light in water is less than speed in air confirming the wave nature of light . Wave theory was not firstly accepted because of great authority of Newton . However Thomas Young in 1801 performed his famous interference experiment firmly establishing the wave theory .

The phenomena of interference , diffraction ,reflection and refraction are well explained by the wave model of light . The only difficulty is that for a wave to propage a medium is needed. But light can travel in vaccum also . This was explained by Maxwell when he proved the electromagnetic nature of light .Maxwell developed a wave equation by assuming light is a propagation electric and magnetic vector mutually perpendicular to each other.From this he calculated the speed of light in vacuum . Value so obtained is very close to the experimentally obtained value of speed of light . So he confirmed light as electromagnetic wave able to propagate on free space .Light waves are associated with changing electric and magnetic field .Changing electric field produces time and space varying magnetic field and changing magnetic field produces time and space varying electric field .The changing electric and magnetic field result in propagation of electromagnetic waves in vacuum .

Huygens Principle : 

When we drop a small stone on the surface of calm pond ,waves are generated and spread out from the point of impact .Waves are seen as circular ring spreading out from point of origin . All points on such a circle are oscillating in phase because there are at same distance from the source . So locus of all points oscillating in phase is called wave front . A wave front is defined as surface of constant phase .The speed with which the wavefront moves outwards from the source is called speed of the wave . The energy of the wave travels in a direction perpendicular to the wave front . If the wave front is a sphere then it is called spherical wavefront . At a large distance a portion of the sphere can be considered as plane and we get plane wave front .

If we know the shape of wavefront at a time t= 0 we can construct the wave front at later time using Huygens principle .It is essentially a geometrical construction. According to Huygens principle , each point on the wavefront is the source of secondary waves and the  secondary wavelets generating from these points spread out in all directions with the speed of the wave .If we draw a common tangent to all these spheres we get the position of wavefront at later time .

If we want to draw the shape of wavefront at time t , we draw spheres with radius vt  at each point on the wavefront( t =0) and if we draw a common tangent to all the spheres we get spherical wavefront at time t .The new wavefront G1G2 is again a spherical wavefront. The above model has a short coming , there is a backward wave .Huygens argued that amplitude of backward wave is zero .

With use of Huygens principle we can construct plane wavefronts also 

Refraction of a plane wave : 

We can derive laws of refraction using Huygens principle .

We will now use Huygens principle to derive laws of refraction.Let v1 and v2  are the speed of light in medium 1 and 2 respectively.We assume a plane wavefront A'A incident on the interface at an angle I as shown in the figure. Let t be the time taken by the wavefront to travel the distance BC . Thus BC= v1t . In order to determine the shape of refracted wavefront we draw a sphere of radius v2t  from the point A .Let CE represent a tangent plane drawn from point C to the sphere . Then AE = v2 t and CE is the refracted wave front . From the triangle ABC and AEC we readily get , 

Sin i = BC/AC = v1t/AC 

Sin r = AE/AC = v2t/AC , where i and r are the angle of incidence and refraction respectively . From the above relations we get ,

Sin I/Sin r = v1/v2 

From the above relation we get the result , if i > r then then v2< v1 implies if velocity is less in second medium rays bend towards the normal . If experimentally proved contrasting the corpuscular theory of light . Now , n1 = C/v1 ( n1 is the refractive index of first medium) Similarly for the 2nd medium 

n2 = C/v2  So

 n2/n1 = v1/v2

Giving Sin i /Sinr = n2/n1

Or , n1 Sin i = n2 Sin r , this is Snell's law of refraction.

If we take BC = one  wavelength in medium 1(lamda1) then AE will also be one wave length(lamda2) in medium 2 

So v1/v2 = lamda1 /lamda2

This implies v1/lamda1 = v2/lamda2

Gives the frequency is same in both the media .

So when light travels from one to another medium its wavelength is changed but not its frequency .

Reflection of a plane wave front by a plane surfac : 

We consider a plane wavefront AB incident at an angle I on a reflecting surfac MN . If v represents the speed of light and t is the time of taken by the  wavefront to reach from B to C then the distanc BC = v t In order to construct the reflected wavefront  we draw a circle with radius v t  from the point A . CE represents the tangent to the circle from C . Obviously AE = BC = v t 

. The triangle AEC and ABC  are congruent.Therefore i = r . This is law of reflection.

With the laws of reflection and refraction we can understand behaviour of lenses ,prism and mirror . When a plane wavefront passes through a thin prism emergent wavefront remain plane with a tilt as shown in the figure .But a plane wavefront passing through a lens it becomes spherical with radius equal to focal length and when reflected by a concave mirror it also becomes spherical wavefront with radius R/2 .

Coherent and Incoherent Addition of waves : 

Here we will discuss about interference pattern produced by superposition of two waves . The superposition principle :At a particular point in a medium the resultant displacement produced by a number of waves is the vector sum of displacements produced by each of the waves . 

We consider two niddles on the surface of water moving up and down periodically and identically .They produce two  water waves and at a particular point the phase difference produced by the two waves remain constant . When this happen we say the sources are coherent .We consider the points where S1P = S2P .Since the distances of the point P is same from both the points The two waves will reach the point P at the same times . So  the two displacement can be represented by ,

y1 = a cos wt  

and  y2 = a cos wt 

So the resultant displacement y is ,

y = y1 + y2 = 2a cos wt , so the resultant intensity is given by ,

I = 4 Io , Io is the intensity produced by individual waves and it is proportional to a^2 , amplitude square . Here the two waves interfere constructively and what we call it as constructive interference .

If the path difference at point P of waves coming from the two sources is 2 x wave length then the the waves can be written as

 y1 = a cos wt 

and y2 = a cos (wt - 4 pi)= a cos wt  , where path difference of 2 x wave length implies a phase difference of 4 pi . The two waves are once again in phase so the resultant intensity is 4 Io giving constructive interference .

But if the path difference is 2.5 x wavelength then the waves may be written as y1 = a cos wt and

y2 = a cos ( wt + 5 pi) = - a cos wt 

The two waves are out of phase calcelling each other giving intensity zero. Here the waves interfere destructively .

To summarise: If we have two coherent sources S1 and S2 vibrating in phase and for arbitrary point P where the path difference, 

S1P - S2P = n x lamda ( n = 0 , 1 ,2, 3 .......) we will have constructive interference  and the resulttant intensity will be 4 Io .

On the otherhand if the path difference at P is (n+1/2) multiple of wavelength i.e 

S1P - S2P = ( n +1/2)x lamda  ( n =0,  1 , 2 , 3 ........) then the waves will interfere destructively giving intensity zero at the point P.

For any arbitrary point G let the difference between two waves is ዋ .The can be represented by ,

y1 = a cos ω t and y2 = a cos( ധt + φ)

And the resultant displacement is given by , 

y = y1 + y2 

    = a [ cos ωt + cos( ω t + φ) 

    = 2 a cos( φ /2) cos ( ω t + φ/2)

The amplitude is given by 2 a cos (φ/2) and therefore intensity is given by 

I = 4 Io cos^2( φ/2) 

If  φ = 0 , 土 2π ,土4 π,  ........

We have constructive interference.

On the other hand if 

φ = 土 π, 土 3 π, 土 5 π .... We have destructive interference leading to zero intensity .

Now if the two sources are coherent the phsase difference fi will not change with time at any  point i.e maxima and minima will not change with time and we will get stable interference pattern . If the phase difference of the two sources changes rapidly with time we say the sources are incoherent. This is what happens when two separate sources illumanate a wall .

geometrical optics

Introduction: 

Electromagnetic radiation belonging to the wave length range between 400 nm to 800 nm is called light . It travels with an enormous speed in straight line in vacuum . It's speed in vacuum is 3 x 10^8 m/s . It's speed in vacuum  is the highest speed attainable in nature . A light wave can be considered to travel from one point to another in a  straight line joining them . Such a path of light is called ray of light . A bundle of light rays is called a beam of light .We will discuss the phenomena of reflection refraction and dispersion of light using ray picture. Next we understand some optical instruments such as microscope , telescope and human eye using the ray property of light .

Reflection on spherical surface : 

We know the laws of reflection : 1) Incident ray , reflected ray and normal on  the reflecting surface at the point of incident all lie in the same plane .2) Angle of incident ( angle between incident ray and normal ) and angle of reflection ( angle between reflected ray and normal ) are equal to each other . These laws are valid in all reflecting surfaces whether it is plane or curved .In curved surface normal is drawn on the tangent at the incident point  and it passes through the centre of curvature of the curved surface. Centre of curvature is the centre point of the sphere forming the curved mirror .

The geometric centre of a spherical mirror is called its pole . In case of lens it is called optical centre.The line joining between pole and centre of curvature is called principal axis . In case of lens principal axis is the line joining between optical centre and principal focus .

Sign convention: 

In Cartesian sign convention here pole is taken as origin and the distances measured in the direction of light rays are taken as positive and the direction measured against the light rays is taken negative . The distance upward to principal axis is taken positive and the downward to the principal axis is assumed negative . 

Focul length of spherical mirror: 

Light rays travelling parallel to the principal axis after reflection from spherical  mirror converge to a point on the principal axis for concave mirror and for convex mirror light rays appears to diverge from a point on the principal axis . This point on the principal axis is called Principal focus . The distance between the pole and focal point is called focal length of the mirror.

Focal length is denoted by f and radius of curvature is R . Let us derive the relation between them . 

Let a light ray parallel to the principal axis incident on M .After incident on it reflects and passes through the point F on the principal axis . The normal at M passes through the centre of curvature C .Let MD is perpendicular drawn on principal axis . Since the angle of incidence is equal to angle of reflection from the triangle CMD  we get MD/CD = tan(theta) .and triangle FMD,  MD/FD = tan(2theta) Since angle theta is small we can write  theta = MD/CD and 2 theta = MD/FD . From these relation we get 2 MD/CD = MD/FD Or CD = 2FD . D is very close to O . so we ca write CO = 2OF or R = 2 f 

The mirror equation : 

If rays emanating from a point actually meet at another point after reflection or refraction then that point is called the image of the first point .The image is real if the rays converge to the point and image is virtual if rays appear to diverge from the point . 

In principle any two rays emanating from a point find their point of intersection and thus get the image of the point due to reflection from a spherical mirror . To draw the image it is convenient to follow any of the two procedures from the following.

i) The ray from the point coming para to the principal axis after reflection goes through the focus of the mirror.

ii) The ray passing through the centre of curvature of the concave mirror retraces back to its path after reflection.

iii) Any ray passing through the focus of the mirror becomes parallel to the principal axis after reflection.

iv) The ray incident at any angle at the pole is reflected following the law of reflection.

Figure shows the rays diagram for the above  three rays . Here AB is image and A'B' is real image .The two right angled triangles A'B'F and MPF are similar .Therefore,

A'B'/PM = B'F/PF ( MP is considered perpendicular to principal axis for small mirror) 

Or A'B'/AB = B'F/PF ( PM = AB)

Since  angle APB' = angle A'PB so triangle ABP and A'PB' are similar .

Hence 

AB/A'B' = PB/PB'  

From the above two relations we get ,

B'F/PF = PB'/PB 

Or, (B'P - FP)/PF = PB'/PB

Or ,( - v + f )/-f  = - v/-u 

When simplified we get 

1/v + 1/u = 1/f 

This is known as mirror equation.

Magnification : 

It is defined as the ratio of image height and the object height= A'B' / AB = h'/h  From the above relation we get A'B'/ AB = PB'/PB = v/u 

So magnification, m = -h'/h = -v/u ( image is inverted so the negative sign ) 

REFRACTION : 

When a light rays travelling from one medium incidents obliquely on the interface of another medium a part of the light ray transmit with direction changed  at the interface of two medium . It is called refraction of light . Snell experimentally obtained the following two laws of refraction .

I) The incident ray reflected rays and the normal to the interface at the point of incidence all lie in the same plane .

II) The ratio of sine of angle of incidence to the sine of angle of refraction is constant for a given pair of media and for a given colour of light .

We have  sini/son r = n21, where n21 is a constant and called the refractive index of medium 2 with respect to medium 1 .It is Snall's law of refraction. If n21 is greater than 1 then medium 2 is optically deser medium .If n21 is less than unity theefium two is rarer medium . If n21 > 1 then  i > r reverse is the case if  n21<1 . Optical density bshould not be confused with mass density. Optical density may be greater where mass density is lower . In case of water and terpenpine oil .mass density of water is greater than terpentine oil. But optical density of terpentine oil is greater than water .

Since the path of light is reversible, n21 = n12 , where n12 is refractive index ofedium 1 wth respect to medium 2 .

It can also be easily proved , 

n23 = n21 x n13  

From the laws of refraction it can be proved that when a ray of light pass through a rectangular glass slab the incident ray and emergent ray are parallel. Another familiar example is : the bottom of tank filled with water appears to be raised . It can shown that apparent depth h1 is real depth h divided by refractive index of the medium . Proof : n = sin I /sin r =  tan i/ tanr ( i, r are small )= h2/h1 ( h2 real depth , h1 apparent depth ) 

Total Internal Reflection: 

When light travel from an optically denser medium to rarer medium at the interface light is partly reflected back to the 1st medium .it is called internal reflection . But the refractive angle is greater than incident angle i.e retracted ray bends away from the normal .If we increase the incident angle at a particular value of it refractive angle becomes 90 degree. Any further increase of refractive angle is not possible. This angle of incidence for which refractive angle is 90 degree is called critical angle of the two media .If incident angle is increased greater than critical angle light will get reflected totally from the interface between the two medium .This reflection is Total Internal Reflection.

In the figure above anglevAO3N is critical angle (ic) for the given pair of media . 

So ,Sinic = n21

Hence n12 = 1/Sinic 

Refractive index and critical angle of some media is given below :

Water  =  refractive index 1.33 Critical angle 48.75 degree 

Crown glass = 1.54 , 41.14 

Denser flint glass = 1.62 , 37.31 

Diamond = 2.42 , 24.41

A demonstration for total Internal  Reflection: 

Take a glass beaker with clear water in it . Add a few drops of milk to make it a little turbid .Take a laser pointer and shine it's beam through the turbid water .The path of the last light rays can be seen inside the water .Shine the beam from below the beaker and it strikes the upper layer of water . We can see the beam is divided into two parts at the surface, one is transmitted to the air and the other is reflected back into the water. If we increase the incident angle gradually we find a value of incident angle there is no rays transmitted into the air ,the total rays is reflected into the water . This is total Internal Reflection. 

Total internal reflection in nature and it's technological applications : 

i) 

Prism : prism designed to bend light by 90 degree or 180 degree make use of total internal reflection.Such a prism is used to invert images without changing their shapes .In this case critical angle of the glass making the prism must be less than 45 degree .

ii) Optical fibres : Optical fibres are extensively used to transmit audio and vedio signal for long distance using the total internal reflection. Each fibre consists of two parts one is core and the external part is called cladding .The refractive index of the core material is greater than that of cladding . When a signal in the form of light is directed at one end of the fibre at a suitable angle it undergoes repeated total internal reflections along the length of the fibre and finally comes out of the other end . Since the rays undergoes total internal reflection there is no appreciabke loss of intensity of the light signal . Optical fibres are also used as 'light pipes' to facilitate visual examination of internal organs like stomach and intestines . In decorative lamp optical fibres are used.One end of the fibre is held at the light source and from the other end we get a dot of light .

Refraction at Spherical Surfacesand by  Lenses : 

So far we considered only refraction at plane interface . We shall now consider refraction at Spherical interface seprating between two transparent media . An infinitesimal part of a spherical surface can be considered as plane surface and the law of reflection hold good . The normal at the point of incidence is perpendicular to the tangent plane drawn at the point of incidence. First we consider refraction by a single spherical surface and then follow it by thin lenses .A thin lens is a transparent medium bounded by two surfaces , at least one of which should be spherical .Applying the formula of image formation by a single surface to two surfaces of a lens we will get lens maker formula .

Refraction at Spherical surface : 

In the figure O is a point on the principal axis and I is it's image and R is radius of curvature.The rays are incident from a medium of refractive index n1 to another of refractive index n2 .We take the aparture of the surface is small compared to other distances involved. NM will be taken perpendicular to principal axis .For small angles ,

 tanNOM = MN/OM 

and tan NCM = MN/MC 

tan NIM= MN/MI 

For triangle NOC,  i is exterior angle 

Hence , i = angle NOM + angle NCM 

i = MN/OM + MN/MC 

Similarly , r = angle NCM - angle NIM 

or, r = MN/MC - MN/MI 

By Snell's law 

n1 sini  = n2 sin r 

For small angles , n1 i = n2 r 

n1( MN/OM + MN/MC) = n2(MN/MC - MN/MI) 

Or, n1/OM + n2/MI = (n2 - n1)/MC

Now taking sign convention ,

OM = - u , MI = v , MC = R 

So we have ,

n2/v - n1/u = ( n2 - n1)/R 

u is object distance , v is image distance R is radius of curvature.n1 and n2 are refractive indices of 1 and 2 medium respectively.

Refraction at Convex lens : 

The image formation can be seen in terms of two steps .

The first refracting surface  forms the image I1 of the object O .The image I1 acts as a virtual object for the second surface that forms the image I .Applying the formula for refraction at Spherical surface to the first surface ABC we get , 

n1/OB + n2/BI1 = (n2 -n1)/BC1

A similar procedure applied to the second interface  ADC we get ,

-n2/DI1 + n1/DI = (n2 - n1)/DC2 

For a thin lens BI1 = DI1 and adding the above two equations , 

n1/OB + n1/DI = (n2 -n1)(1/BC1 + 1/DC2) 

If  the object is taken to infinity then OB tends to infinity and DI = f  the above equation gives ,

n1/f = (n2 - n1)( 1/ BC1 + 1/DC2) 

Now BC1 = + R1 and DC2 = -R2 

So the equation becomes 

1/f =( n21 - 1)( 1/R1 - 1/R2) as n21 = n2/n1 

The above equation is known as lens maker's formula .

The formula is equally valid for concave lens also .In concave lens R1 is nagative , R2  positive giving f negative .

Derivation of thin lens formula : 

Considering the refraction at two spherical surfaces , we have the equation : 

n1/OB + n1/DI  = ( n2 - n1)( 1/ BC1 + 1/BC2) 

Or 1/OB + 1 /DI = (n21 - 1)(1/R1 - 1/R2 ) ( OB is object distance = -u, DI is image distance = v and using lens maker formula )

Or , 1/-u + 1/v = 1/f 

Or, 1/v - 1/u  = 1/f 

It is known as lens formula for thin lens .

Image formation by a lens : 

To locate the image of a point of an object kept on th principal axis we have to follow the path of any two rays emerging from the point and undergoes refraction at the lens and then find the point where the refracted rays meet .This meeting point is called the image of the point .

We may consider any two of the following three rays undergoing refraction. 

i) A ray emanating from the point and parallel to the principal axis after refraction passes through the second focus F' 

ii) A ray passing through the optical centre passes undeviated after refraction.

iii) A ray emerging from the object and passing through the 1st focus emerges parallel to the principal axis

 In case of concave lens the rays will not meet after refraction but they seem to come out from a point .

Power of a lens :

The power of a lens is a measure of convergence or divergence of a lens when light falling on it . The power P is the tangent of the angle by which it is converged or diverged when light falling parallel to principal axis falling at a unit distance from optical centre .

tan (delta) = h/f ; if h = 1, then tan(delta)= 1/f 

Delta= 1/f 

Delta = 1/f for small values of delta 

P = 1/f 

The SI unit of power 1D = 1 m^-1 

The power of 1 metre focal length lens is 1 doipotre . It is positive for converging lens and negative for divergence lens .

Combination of thin lenses in contact : 

we consider two lenses A and B of focal length f1 and f2 placed in contact with each other .Let an object is placed at O . The lens A produced an image of I1 . It serves as object to the lens B and final image is formed at I .Let the object distance is u and image distance for I1 is v1 and final image I is formed at a distance v from the lens combination. From the image formed by the first lens A we have , 

1/v1 - 1/u = 1/f1 

For the image formed by 2nd lens B we have , 

1/v -1/v1 =  1/f2 

From the above two relations we get ,

1/v - 1/u = 1/f1 + 1/f2 ,

From the above relation we can say that if the focal length of the combination is f then 

1/f = 1/f1 + 1/f2 

Here two lenses can be regarded as a single lens with focal length f 

If there are more than two lenses the formula can be written as 

1/f = 1/f1 +1/f2 + 1/f3 + ........

So the power of the equivalent lens is P = P1 +P2 +P3 +.........

If the individual magnification is m1 ,m2,m3 etc then total magnification m is given by 

m = m1 m2 m3 .......

Such a combination is used to design lens for cameras, microscope and telescope etc 

Refraction through a prism : 

The picture above shows the refraction of a light rays through a prism . The angle of reflection incidence at the first surface is i and r1 for the second surface the corresponding angles are r2 and e .The angle between incident rays and emergent ray is delta ( ∆) 

In the quadrilateral AQNR , the angles at Q and R are right angles .So the sum of other angles is 180 degree 

Angle A + angle QNR = 180 degree 

From the triangle QNR , r1 + r2 + QNR = 180 degree 

Comparing the above two equations 

A = r1 +r2 

The total deviation delta is the sum of deviations at two surfaces 

∆ = ( i - r1 ) + ( e - r2) 

Thus ∆ = i + e - A 

Thus angle of deviation depends on the angle of incidence i . For any value of delta we have two values i and  e since path of light is reversible. 

At the minimum deviation i = e and hence r1 = r2 .So we have at minimum deviation 

2r = A or r= A/2 

So Dm = 2i - A 

Or i = ( A + Dm)/2 

So the refractive index of the prism 

n21 

= n2/n1 

= sin[(A + Dm)/2]/sin[A/2]

= (A + Dm)/2÷ A/2 ( for angles ,thin prism )

Dm = (n21 - 1) / A 

It means prism do not deviate light much .

Optical Instruments : 

A number of optical devices are designed utilising the property of refraction and reflection. Microscope , telescope and binocular are formed using the above properties of light .

The Microscope : 

A simple microscope is converging lens with small focal length . The lens is held near the object and the eye is positioned close to the lens on the other side .An erect virtual image is formed at a distance so that it can be seen comfortably i.e 25 cm more .If the object is placed at f the image is at infinity.Although if the image is formed at D = 25 cm for comfortable viewing it causes some strain on the eyes . Therefore for relaxed eye comfortable viewing distance is infinity . The linear magnification, 

m = v/u  = v( 1/v - 1/f) = ( 1 - v/f) .

If v= D , for clear vision and using the sign convention, 

m = ( 1 + D/f) , here D is about 25 cm . magnification can also be written as m = h'/h where h' is image size ,and h is object size 

If ,theta is the angle subtanded by object kept at D then 

tan 'theta' = h/D = theta, for small angle . 

This is one less than the magnification when the image is at near point .

In discussion of microscope and telescope we assume image is at infinity.

A simple microscope has a limited maximum magnification (< 9 ) for realistic focal length . For much larger magnification one uses two lenses compounding the effect .This is known as compound microscope.

The lens nearest to the object called objective form real inverted  magnified image .This serves as object for the second lens called eye piece that functions as simple microscope producing enlarge virtual image .

β

magnification produced by objective m zero = h'/h = L/fo .

magnification produced by eyepiece me = 1 + D/fe 

For infinity focusing 

me = D/fe 

The final magnification,

m = mo me =( L/fo) (D/fe)

 Telescope: 

The telescope is used to see distant objects and it provides angular magnification. The angular magnification can be written as the ratio of  angle subtainded by image and the angle that makes by the object at the objective .

So , m = mo .me 

Or , m = tan β/ tan α = h/fe ÷h/fo 

Hence ,m = fo/fe 

Terrestrial telescope has in addition a pair of investing lens to make the final image erect. Refracting telescope can be used as both astronomical and terrestrial observation.

The main considerations with astronomical telescope are its light gathering power and its resolving power .The former is clearly depends on area of objective .With larger diameter of objective fainter objects can be seen .The resolving power is the ability to observe two distinct separated by small angular separation . it also depends on diameter of objective . But it is very difficult and expensive to make very large size lens that are free from also chromatic aberration and distorsion .For these reason modern telescope use concave mirror objective. Such telescope is called reflecting telescope .

One problem of the reflecting telescope is that it focuses light inside the telescope tube . One must have an observer and eye piece there obstructing some light .The solution to the problem is to deflect light bring focused by another convex mirror and the reflecting light passes through a hole in the objective.

MCQ for H.S

 1. A chemical cell of emf  E and internal resistance r is connected to a resistance R .The potential difference across R is given by : 

A) V = Er/( r + R)  B) V = ER/(R+r)  c)  E(R +r)/2Rr  D) 2ERr/(R+r) 

2.On shunting a galvanometer its : 

A) Current measuring range decreases but current sensitivity increases .

B) Current measuring range and sensitivity  both decreases 

c) Current measuring range and sensitivity both increases . 

D) Current measuring range increases and sensitivity decreases 

3. Which of the following nuclei has lest possibility of undergoing nuclear fission or fusion ? 

A) ₁H²  B)  ₉₂U²³⁵ C) ₂₆Fe⁵⁶ D) ₉₄Pu²³⁹ 

4.Which of the following optical media has the greatest refractive index ?

A) Diamond  B) Glass  C) Water  D) Air 

5.An electric dipole is kept in a uniform electric field such that  its dipole moment p is anti parallel to electric electric field E . The torque acting on the dipole will be : 

A) -2pE  B)) - p E  C) Zero  D) p E 

 6.A 8µF capacitor is charged with 400 V and then its plates are connected to a resistance of 10 kΩ. The heat produced in the resistance is : 

A) 0.16J  B) 0.32J  C) 0.64 J  D) 1.28 J 

7. A body of mass 5kg at rest is raised to a height of 10 m under a constant force of 170 N . At the height of 10 m the velocity of the body will be : (g = 9.8 m/s² ) 

A) 9.8 m/s  B) 15.0 m/s  C) 22.0 m/s  D) 37.0 m/s 

8. A potentiometer wire is 10 m long and has a resistance of 40 Ω. It is connected to a resistance of 20 Ω and an ideal battery of 3 V emf in series . The potential gradient along the wire is : 

A) 0.1V/m  B) 0.2 V/m C) 0.3V/m  D)  0.4 V/m 

9. A spring of length l and force constant k is divided in two parts A and B such that lₐ : lᵦ = 2 : 3 .The force constant of part A will be : 

A) 3/2 k  B) 5/2 k  C) 2/5 k  D) k 

10. Outside a nucleus : A) a proton is stable  B) a neutron is stable  C) neutron and proton both are stable  D) neither proton nor neutron is stable . 

11.  Photons of energy 7.0 eV incident on a surface of work function 2.0 eV . The maximum velocity of the emitted electron is : 

A) 1.33 x 10⁶ m/s B) 5.0 x 10⁷ m/s C) 9.0 x 10⁶ m/s D) 1.4 x 10⁵ m/s 

12.The de-Broglie  wavelength associated with a particle of mass m and kinetic energy E is proportional to : A) m √E  B) √E/m  C) 1/ ✓Em D) ✓mE 

13. Two waves having intensities in the ratio 9 : 1 exhibit interference . The ratio of maximum to the minimum intensities in the interference pattern is : 

A) 10:8  B) 9 :3  C) 4 :1  D) 2 : 1 

14.Which of the following phenomena is not common to sound and light wave ? 

A) Interference  B) Diffraction   C) Reflection  D) Polarization 

15. A circular coil A has a radius r and carries a current I . Another coil with radius 2 r and carries current 2 I . The ratio of the magnetic field Ba/Bᵦ at the centres of the two coils is : 

A) 1 :1  B) 2 :1  C) 4 : 1 D) 1; 4 

16. Which of the following is not an unit of energy  ?

A) Electron -volt  B) Killowatt-hour  C) Joule  D ) Horse power  

17.Complete the sentence given below using the options provided in the bracket .

( kinetic energy , total energy , sum of mass and energy , potential energy ) 

i) The quantities conserved in elastic  collision are linear momentum and ------------.

ii) In nuclear reactions ----------- is conserved . 

18. A spherical metal bob of mass 100 g is attached to 1 m long string and suspended from a rigid support . It is  then displaced to  one side from its equilibrium position O to a position A which is 5 cm above its equilibrium position ( g = 10 m/s² ) 

i ) The potential energy of the bob at position A is 

A) 5000 J B) 50 J  C) 0.5 J  D) 0.05 J

ii) When released from position A with what velocity will it pass through position O ? 

A) 1 m/s  B) 10 m/s  C) 100m/s  D) 1000 m/s 

19. Read the passage given below and the questions that follow : 

The force(F) vs displacement(x) of a body is given in the above fig. fill in the blanks given below from the options provided .

I) The body starts decelerating from the point ------------- of the graph .

A) B. B) C   C ) D  D)  E

ii) The minimum work is done when it moves between ---------------.

A) 2 m - 3 m. B) 3 m - 4 m C) 4 m - 5 m D) 1 m - 2 m 

VSA for MP2025

 1.Answer the following questions ( VSA) :

1.1 Fill in the blank : In Troposphere temperature ---------- with increase in altitude .

1.2 Mention one harmful effect of global warming .

1.3 Which atom released from chloroflorocarbon under the influence of UV rays decompose ozone into oxygen ?

1.4 Mention one use of bio- gas .

1.5 What is the role NO in decomposition of ozone in the ozone layer ?

1.6 Among charcoal , petrol and ethanol which one is a fossil fuel ?

1.7 Write down the unit of calorific value of fuel .

1.8 Does the temperature increase or decrease with increase of altitude in the Stratosphere ?

1.9 Which radiation , coming from the sun , is prevented by the Ozone layer from falling on the earth’s surface ?

1.10 Which fuel gas is harvested from coal bed ?

1.11 Name a gas present in air , the increase in amount of which causes global warming .

1.12 Name an energy source which can be used for sustainable development.

1.13. Mention one fuel which is alternative to fossil fuel .

1.14 Write the name of a gas which increases atmospheric temperature .

1.15 Fill in the blank: -------- radiation ,coming from the sun is prevented by the ozone layer from falling on the earths surface .

1.16 Name a gas present in the atmosphere which is not a greenhouse gas .

1.17 What is meant by – the calorific value of a sample of coal is 30000 kJ/kg ?

1.18 Why can wind energy be used for sustainable growth development ?

1.19 Write whether the following statement is true and false : Inder the same condition of temperature and pressure equal volume of CO2 and N2 gas contain different number of molecules.

1.20 Write down the relation between the pressure and volume of a given mass of a gas according to Boyle’s law when temperature remains constant.

1.21 What is the unit of M in the equation PV = W/M RT ? ( Symbol s have usual meaning)

1.22 Under constant pressure, at what temperature in degree Celsius the volume of an ideal gas will be zero according to Charles ‘ law ?

1.23 State whether the following statement is true or false : The volume of gas molecules is taken into consideration in Avogadro’s Law .

1.24 The product of volume and pressure of how many gram of N2 gas is 224 litre atmosphere at STP ?(N = 14)

1.25 Write whether the following statement is true or false : The speed of the gas molecules contained in a closed vessel at fixed temperature and pressure is the same .

1.26 What is the nature of V versus T graph according to Charles’ law .

1.27 What is the SI unit of pressure of gas ?

1.28 What are the constants in Charles law ?

1.29 Write whether the following statement is true or false : Per degree interval in Kelvin scale of temperature is equal to perform degree interval in Celsius scale .

1.30 At a certain pressure the volume of a fixed mass of a gas at a temperature of 00 C is V0 ; what will be the increase in volume of the gas , according to Charles’ law , if the temperature of the gas is raised by 10C keeping the pressure unchanged ?

1.31 What will be the ratio of molecules present in 1 L of H2 and 4 L CO2 gas at STP ?

1.32 Between apparent and real expansion coefficients of a liquid which one is it’s own characteristic?

1.33 What is the SI unit of thermal conductivity ?

1 34 Whether the following statement is true or false? The real expansion of any liquid depends on the expansion of the vessel in which it is kept .

1.35:Among iron , invar and copper which one has the least co-efficient of linear expansion?

1.36 State whether the following statement is true or false. The constituent of a material change position during conduction of heat through it .

1.37 The width and the cross- section of a conductor remaining unchanged , what is the relation between the thermal resistance and thermal conductivity of that conductor?

1.38 Write whether the following statement is true or false: Among copper , invar and iron the linear expansion coefficient of iron is the lowest .

1.39 What is the unit of volume expansion coefficient?

1.40 Write whether the following statement is true or false: The value of coefficient of linear expansion of solid is the same in Celsius and Kelvin scale .

1.41 Arrange diamond iron and silver in the order of decreasing thermal conductivity.

1.42 Give a natural example of dispersion of light .

1.43 Which type of lens can rectify myopic vision ?

1.44 Between the angle of incidence and the angle of refraction which one is greater when light travels from a rarer to a denser medium ?

1.45 What type of mirror is used in the viewfinder of a motor car ?

1.46 What will be the angle of incidence when a ray of light passes through the centre of curvature of a concave mirror ?

1.47 How many rectangular surfaces are there in a prism ?

1.48 What is meant by the pole of a spherical mirror ?

1.49 Write down one use of X- ray .

1.50 Is it possible to form a virtual image of shorter length of the object by any mirror ?

1.51 What type of mirror is used in the head light of a motor car ?

1.52 If the refractive indices of a medium for red and blue colors of light are nr and nb respectively , which one is greater in magnitude ?

1.53 Write one use of convex mirror?

1.54 A ray parallel to the principal axis of a concave mirror follows which path after reflection by the concave mirror ?

1.55 How much work has to be done when 1 coulomb of charge is taken against a potential difference of 1 volt ?

1.56 From which part of a radioactive atom is beta particle emitted ?

1.57 what is the atomic number of the daughter element produced if an alpha particle is emitted from 238 U92 ?

1.58 How does the resistance of a semiconductor change with increase of temperature?

1.59 Which type of energy is transformed to electrical energy in a dynamo ?

1.60 Arrange alpha, beta and gamma rays in ascending order of their penetrating power ?

1.61 Which kind of nuclear reaction is the source of sun’s energy ?

1.62 Give an example of semiconductor.

1.63 A thin wire and a thick wire of the same conducting material have the same length .Which one of them will carry more current when connected to the same potential difference?

1.64 Mention one misuse of nuclear fission reaction .

1.65 Which law explain the release of huge amount of energy in nuclear fusion ?

1.66 Write down one use of X - ray .

1.67 Name a machine where electrical energy is converted to mechanical energy.

1.68 Apart from the live wire , what are the two other wires in the household circuit ?

1.69 What is the SI unit of electric charge ?

1.70 What is the unit of electrical conductance ?

1.71 What are the constituents of fuse wire ?

1.72 Kilowatt-hour is the unit of which physical quantity ?

1.73 If the same potential difference is applied between the two ends of iron and copper wires having the same length and cross section , does equal amount of current flow through the two wires ?

1.74 Draw I – V graph according to ohm’s law in the case of metallic wires .

1.75 Mention one peaceful use of atomic energy.

1.76 Fill in the blank : gamma ray is ------------ wave of short wavelength .

Mp 2025MCQ

 MCQ for MP 2025

1. Choose the correct alternative (MCQ) :answer keys for Q1 to 20 :(https://forms.gle/Qn2ZzzjugwYt7Vt18)  Q 21 to40 : (https://forms.gle/AK3F2oryd2T7dH6A6)

1.1 Which of the following is not a greenhouse gas ? a) methane b) water vapour c) carbon dioxide d) oxygen

1.2 Which of the following greenhouse gases has a maximum contribution towards global warming? a) N2O b) CH4 C) CO2 d) H2O vapour

1.3 Which among the following gases absorb long wavelength infrared radiation emitted from the earth’s surface ? a) N2 b) O2 c) CH4 d) He

1.4 Which among the following gases does not help in the depletion of ozone in the ozone layer ? a) NO b) NO2 c) CFC d) CO2

1.5 Which layer of atmosphere has the greatest density ? a) Troposphere b) Stratosphere c) Mesosphere d) Thermosphere

1.6 Which of the following gas helps in the depletion of ozone layer? a) CO2 b) Ar c) CFC d) He

1.7 Which is the main constituent of bio gas ? a) CH4 b) CFC c) CO2 d) CO

1.8 When do real gases behaves as ideal gas ? a) at high temperatures and high pressure b) at high temperatures and low pressure c) at low temperatures and low pressure d) at low temperatures and high pressure

1.9 At STP number of moles in 44 g CO2 is – a) 1 b) 2 c) 3 d) 4

1.10 What is value of PV for 4 g of H2 has at STP ( H = 1 ) a) RT b) 2 RT c) 4 RT d) 0.5 RT

1.11 In case of ideal gas ratio of coefficient of volume and pressure expansion is – a) 0 b) 1 c) 1/273 d) 2

1.12 The P – V graph of Boyle’s Law related to gas is –

1.13 What is the value PV for 11.2 litres of an ideal gas at STP ? a) 2RT b) RT c) 0.5 RT d) 11.2 RT

1.14 At STP 2.24 L is occupied by – a) 4.4 g CO2 b) 0.64 g SO2 c) 28 g CO d) 16 g O2 ( C = 12 , O = 16 , S = 32 )

1.15 According to Boyl’s law which is the PV – P graph –

1.16 Which of the following is the SI unit of pressure ? a) Nm2 b) Nm-2 c) Nm d) N

1.17 Vapour density of a gas is 32 . Which of the following is its molecular weight ? a) 8.b) 16 c) 32 d) 64

1.18 If the vapour density of a carbon containing gaseous substance is 13 , which of the following can be its molecular formula? a) CO2 b) C2 H4 c) C2H6 d) C2H2

1.19 How many molecules of CO2 will be produced when one mole of C reacts completely with 1 mole O2 ? a) 6.022 x 1023 b) 1.806 x 1024 c) 6.022 x 1022 d) 6.022 x 1024

1.20 According to to the following chemical equation CH4 + 2 O2 = CO2 + 2H2O what volume of O2 will be required to burn 10 mole of CH4 at STP ? a) 448 L b) 224 L c) 44.8 L d) 22.4 L

1.21 The relation between molecular mass ( M) and vapour density ( D) of a gaseous substance is – a) 2M = D b) M = D2 c) M = 2.8 D d) M = 2D

1.22 How many gram of O2 will be required to make CO2 by burning 12 g of C completely ? ( C = 12 O = 16) a) 32 g b) 12 g c) 16 g d) 44 g

1.23 On which of the following does the thermal conductivity of a metal depend ? a) temperature b) length c) nature of the material d) area of cross section.

1.24 The unit of co-efficient of linear expansion of a solid is – a) m b) m-1 c) o C-1 d) oC

1.25 For a solid how many types of thermal expansion co-efficients are there ? a) one b) two c) three d) four

1.26 Which among the following substances has the highest heat conductivity? a) Silver b) Diamond c) Copper d) Aluminium

1.27 How many types of thermal expansion coefficient are there for a liquid ? a) 0 b) 1 c) 2 d) 3

1.28 What is the relation between the radius of curvature and the focal length of a mirror?a) f = 2r b) f = r/2 c) f = r/3 d) f = 3/2 r

1.29 Which of the following colours deviates most when white light is retracted by a prism ? a) red b) yellow c) violet d) green

1.30 From the following which states the relation between refractive index and wavelength of light ?

1.31 An object is placed in between the optical centre and focus of a thin convex lens . What is the nature of image of the object ? a) real and inverted b) virtual and inverted c) real and erect d) virtual and erect

1.32 Magnified virtual image is formed by a) convex mirror b) convex lens c) concave lens d) plain mirror .

1.33 When a ray of light is incident perpendicularly on a transparent glass slab the angle of incidence is – a) 0o b) 1800 c) 30o d) 90o

1.34 Which one of the following has the highest wavelength? a) X- ray b) gamma ray c) infrared ray d) ultraviolet ray

1.35 In case of refraction , if the angle of incidence and the angle of refraction are 450 and 300 respectively, then the angle of deviation is – a) 750 b) 150 c) 7.50 d) 37.50

1.36 If a beam of red light and a beam of violet light are incident at the same angle on the on the inclined surface of a prism from air medium and produce angles of refraction r and b respectively, which of the following is correct? a) r = v b) r = 1/v c) r > v d) r < v

1.37 A point source of light is placed at the centre of curvature of a concave mirror. The angle of deviation of the rays incident from this source and reflected from it , is – a) 00 b) 1800 c) 900 d) 3600

1.38 The wavelength of which of the following is greater than the visible light ? a) X- ray b) infrared ray c) gamma ray d) ultraviolet ray

1.39 Dentists use – a) convex mirror b) convex lens c) concave mirror d) concave lens

1.40 In the case of refraction of white light through prism the light of colour which deviates least is a) yellow b) orange c) red d) violet

1.41 If a reflected ray from a concave mirror makes an angles of 450 with the normal , the angle of incidence will be – a) 900 b) 22.50 c) 1350 d) 450

1.42 Which of the following physical quantities represents ampere ? a) coulomb – second b) volt-1ohm c) volt- ohm-1 d) volt- ohm .

1.43 Resistivity of which of the following decreases with increase of tempe? a) conductor b) semiconductor c) superconductor d) insulator

1.44 Which of the following is the correct order of ionisation power for alpha , beta and gamma rays ? a) alpha>beta >gamma b) alpha > gamma> beta c) gamma>beta>alpha d) beta> alpha> gamma

1.45 Which of the units given below is the SI unit of resistance? a) volt b) ampere c) coulomb d) ohm

1.46 In domestic electric cirçuit the fuse wire is connected to which of the following? a) earth line b) live line c) neutral line d) v th live and neutral line

1.47 Beta – ray emitted from a radioactive element is – a) a stream of electrons b) a stream of protons c) a stream of neutrons d) electromagnetic wave

1.48 Temperature remaining constant if the potential difference between two ends of a conductor is V and the current through the conductor is I , which of the following is true ? a) V proportional to I b) V proportional to I2 c) V proportional to I-1 d) V proportional to I-2

1.49 The relation among electromotive force (V) work(W) and charge (Q) is – a) Q = WV b) Q = V/W c) Q = V/W2 d) Q = W/V

1.50 For the atom produced by beta particle emission from a radioactive atom – a) mass number increases b) atomic number increases c) mass number decreases d) atomic number decreases

1.51 Coulomb’s law related to electric charges applicable when of the two charges – a) one point , one is spherical b) both are spherical c) one is point ,one is extended d) both are points charges

1.52 The characteristics of a fuse wire are – a) high resistance,high melting point b) low resistance, low melting point c) low resistance ,high melting point d) high resistance,low melting point

1.53 Present in alpha particle – a) one proton ,one neutron b) one proton c) two protons ,two neutrons d) one electron

1.54 Charge of an electron is - a) – 3.2 x 10-19 C b) – 1.6 x 10 – 19 C c) 1.6 x 10 – 19 C d) 3.2 x 10 – 19 C

1.55 If the resistance( R) of a conductor and the time of flow of current in the conductor ( t) remain unchanged , the relation between the heat produced (H) in the conductor and flow of current (I) is – a) H proportional to I b) H proportional to 1/I2 c) H proportional to I2 d) H proportional to 1/I

1.56 If 12 C of charge flows through a conductor for 2 minutes the electric current is – a) 6 amperes b) 0.1 amperes c) 24 amperes d) 10 amperes

1.57 What is the unit of conductivity ? a) mho metre-1 b) ohm.metre-1 c) mho.metre d) ohm .metre

1.58 If a current of 0.2 ampere flows through a conductor of resistance 40 ohm , what is the potential difference between the two ends of the conductor? a) 0.5 volt b) 2 volt c) 6 volt d) 8 volt

1.59 The correct order of penetrating power of alpha ,beta and gamma rays is – a) gamma>alpha>beta b) gamma >beta> alpha c) alpha> beta>gamma d) beta> gamma > alpha

1.60 Electric current is flowing through a wire . In this state the wire is elongated in length current will be – a) increased b) decreased c) first increased and then decreased d) will not depend on increase of length

1.61 If 5 ampere current flows through a conductor in 30 second the total charge flowed through the wire is a) 6 coulomb b) 150 coulomb c) 30 coulomb d) 300 coulomb

Mock test math 2024

 Mock Test math 2024.

Class - X Time - 3 hrs 15 min. Full marks - 90 

1. Choose the correct option: 1 x 6 = 6

I) If principal become twice as it's amount in 10 yrs , the rate of simple interest per annum is - 

a. 5% b.10 %  c. 15 % d. 20 % 

ii) If the roots of the equation 3 x² + 8 x + 2 = 0 be p and q , then ( 1/p + 1/q ) is - 

a. -3/8. b. 2/3. c. -4  d. 4 

iii) If the radius of two circles is 3 cm and 4 cm and the distance between their centres is 10 cm then the number of common tangent is - 

a. 0 b. 2 c . 3. d. 4 

iv) If the ratio of curved surfaces of two solid spheres is 16 : 9 , then the ratio of their volumes is - 

a. 64 : 27 b. 4 : 3 c. 27 : 64  d. 3 : 4 

v) The value of ( sin 43º cos 47 º + cos 43 º sin 47 ⁰ ) 

a. 0. b. 1  c. sin 4⁰   d. cos 4 º 

vi) If the mean of the numbers 6 ,7 , 8, x, y, 14 is 9 then - 

a. x + y = 21. b. x + y = 19 c. x -y = 21  d. x -y = 19 

Fill in the blanks( any five): 1 x 5 = 5 

i) If A invests Rs 500 for 9 months and B invests Rs 600 for 5 months in a business then the ratio of their profits is --------------.

ii) If x is proportional to y the x ⁿ ∝ ----------.

iii) The angle in a semi-circle is a --------------.

iv) If the angle of elevation of sun is 45 º ,then the length of shadow and length of post are ----------.

v) Mean ,median and mode are the measure of ---------------.

vi) The number of surfaces of a solid hemisphere is ----------.

3. Write true or false (any five): 1 x 5 = 5

i) On a certain sum for one year the simple and compound interest are the same if interest is calculated annually .

ii) Conjugate of √5 + 2 is √5 - 2 

iii) The volume of a solid hemisphere of radius 2 r is 2/3 π r³ 

iv) The number of common tangent of two circles when they meet externally is 3.

v) The reciprocal of tan (90º - A) is cot A.

vi) Mode is the value of a data having minimum frequency.

Electric Current

                                                                               Electric Current .

We are familiar with electric current from our very childhood .We make use of current in our day to day activities .One layman once told me current is one kind of fire flowing inside wires. Such kind of conception may be  present in many of us. In order to understand what current is we have to understand the basic  structure of atoms . Any type of atom contains  three elementary particles . These are electron proton and  neutron .Electrons and protons are charged particles .electrons are negatively charged and protons  positively charged. Protons reside in the central part of atom and they are not movable .Electron circles round the central part (nucleus) in different shells .In some element electrons revolving in the outermost shells are easily detachable and hence they are moveable from one atom to another . In conductor outer most Electrons become practically free  to move anywhere randomly .These electrons are called free electrons . So in conductor we have some free electrons moving randomly just like random motion of gas molecules  in  a container . When current flows in a conducting wire  another motion known  as drift motion is superimposed on random motion of free electrons due the presence of electric field supplied by battery or other e m f source .Due to drift motion electrons have drift velocity . This drift velocity is not very much only a few centimetre per second . Since electrons are negatively charged they move opposite to the direction of applied field . So to have electric current we need two things one is a substance having free electrons and an electric field  across the substance (wire)  Metals such  as copper ,aluminium   and iron etc are all substances with lot of free electrons .To get electric field we need changing magnetic field as changing magnetic field produces electric fields according to  Farady's law of induction .Another way to get electric field we need static electric charges as in battery .In battery two conducting plates placed slight distance apart with a chemical substance in between them .The plates  become oppositely charged  resulting with a generation of electric field .The current generated with the application of electric field across a conductor is proportional to electric field . This is Ohm’s law .As the electrons flow within wire they collide with ions ,atoms and on themselves . Due to these collisions they transfer energy heating the conductor and they feel difficulty in drifting forward .  So electrons face resistance as they flow inside conductor. This resistance is a property or characteristics of conductor. It depends on length , Cross sectional area and material nature and temperature etc .A good conductor has less resistance . Our body is also a good conductor of electricity as it contains various ions such as sodium ,potassium  etc . When current passes through our body cells will die and death may be it’s  consequent result. A conductor’s  resistance increases with increase of temperature.In  case of insulators resistance is infinite due absence of free electrons .  Insulators such as dry wood ,plastic and glass  have  no free electrons to conduct electricity .But there is a substance called semiconductor that offers resistance in between insulator and conductor.  In semiconductor  with increase of temperature within a certain range resistance decreases .Silicon and germinium are ideal semiconductor .In modern electronic gadgets we find many application of semiconductors . As temperature is lowered to near absolute zero resistance vanishes ie conductor becomes super conductor. In super conductor current is maintained without any loss of power .Here Ohm’s law is no longer a valid law .Current can be maintained without any power source in super conducting wire