`text(Here we will encounter two cases : )`

(a) When object is in denser medium with respect to observer. Here `mu_1 > mu_2`

(b) when object is in rarer medium with respect to observer. Here `mu_1 < mu_2`

From Snell's law

`mu_1sintheta_1=mu_2sintheta_2`

for paraxial rays `theta-> 0`

`=>mu_1theta_1=mu_2theta_2`

`=>mu_1tantheta_1=mu_2tantheta_2`

`=>mu_1 (PQ)/(OP)=mu_2 (PQ)/(IP)`

`=>(IP)/(OP)=(mu_2)/(mu_1)`

Relation between the velocities of object and image if object is moving perpendicular to the surface :

`v_text(O/surface)=d/(dt) (OP)` and `v_text(I/surface)=d/(dt) (IP)`

but `(IP)/(OP)=(mu_2)/(mu_1)`

`=>` `IP=(mu_2)/(mu_1) OP`

differentiating both sides with respect to time we get

`v_text(I/surface)=(mu_2)/(mu_1) v_text(O/surface)`

`text(Shifting of Image : )`

Let an object O be placed at one of the slab (thickness- t, R.l.- `mu`) and the distance between the surface closer to the object and the object be x.
At first surface

`(I^'P)/(OP_1)=mu/1` `=>` `(I^'P_1)=muOP_1=mux`

At second surface

`(IP)/(I^'P_2)=1/mu` `=>` `(IP_2)/(mux+t)=1/mu`

`IP_2=x+t/mu`

Now, shifting will be given by

`Deltax=OP_2-IP_2=(x+t)-(x+t/mu)`

`=>` `Deltax=t(1-1/mu)`

Note that shifting occurs in the direction of propagation of light.

`text(Combination of Slabs : )`

Net shift will be

`(t_1+t_2+t_3+.................+t_n)(1-1/mu_(eq) )=t_1(1-1/mu_1)+t_2(1-1/mu_2)+...........+t_n(1-1/mu_n)`

`=>` `(t_1+t_2+t_3..............+t_n)/mu_(eq)=(t_1)/(mu_1)+(t_2)/(mu_2)+(t_3)/(mu_3).............+(t_n)/(mu_n)`

`text(Shifting in Path : )`

When a ray passes through a slab placed in a medium then after refraction the emergent ray is parallel to the incident ray but it seems that it has translated some distance (`Deltad`, called shifting in path)

In triangle `P_1 P_2 P_4`

`P_1P_2=t/costheta_2`

In triangle `P_1 P_2 P_3`

`Deltad=P_2P_3=P_1P_2sin(theta_1-theta_2)=t/costheta_2 sin(theta_1-theta_2)`

Eliminating `theta_2` (using Snell's Jaw) , we get

`Deltad=t/costheta_2 sin(theta_1-theta_2)=tsintheta_1[1-(mu_1costheta_1)/sqrt(mu_2^2-mu_1^2sin^2theta_1)]`

`text(Bending of an object : )`

When a point object in a denser medium is seen from a rarer medium it appears to be at a depth (`d/mu`). so if a linear object is dipped inclined to the surface of a liquid, (say water) actual depth will be different for its different points and so apparent depth. Due to this the object appears to be inclined from its actual position or BE as shown in figure.

`text(Visibility of two images of an object : )`

When an object is in a glass container and is seen from a level higher than that of liquid in the container as shown in figure, two images `I_1` and `I_2` of object O can be seen simultaneously-one due to refraction at the upper surface while the other at the side surface.

`text(The sun is oval shaped at the time of rising and setting : )`

In the morning or evening, the sun at the horizon and refractive index in the atmosphere of the earth decreases with height. Due to this, light reaching earth's atmosphere from different parts of vertical diameter of the sun enters at different heights in earth's atmosphere and so travels in media of different refractive indices at the same instant and hence bends unequally. Due to this unequal bending of light from vertical diameter, the image of the sun gets distorted and it appears oval and larger. However at noon when the sun is overhead, then due to normal incidence there will be no bending and the sun will appear circular. Similarly you can explain Sun rises before it actually rises and sets after it actually sets.

`text(Stars twinkle : )`

Stars are self -luminous distant object, so only a few rays of light reach the eye through the atmosphere. However, due to fluctuations in refractive index of atmosphere the refraction becomes irregular and the light sometimes reaches the eye and sometimes it does not. This gives rise to twinkling of stars. If from moon or free space we look at a star this effect will not take place and star light will reach the eye continuously.

`text(Tree appear inverted in deserts (mirage) : )`

It is an optical illusion created due to the phenomenon of total internal reflection. This is seen in hot region. In hot areas like deserts surface of earth is very hot. So, air in the lower regions of atmosphere is hot as compared to that in higher regions. This result in variation of density with height and it increases as we go up. In this situation atmosphere can be assumed to be made of large number of thin layers of air.
A beam of light starting from an object say a tree and travelling downward finds itself going from denser to rater medium. Therefore, its angle of incidence at consecutive layers goes on increasing gradually till it surpasses the critical value and is reflected back due to total-internal reflection. A virtual image of the object is seen by eye at E. Due to the disturbance of air, the mirage is wavy in nature, thus giving an illusion for the presence of water which is actually not there. This effect is also called inferior mirage.

`text(Ships appear above in the air in cold countries (looming) : )`

This effect occurs when the density of air decreases much more rapidly with increasing height than it does under normal conditions. This situation sometimes happens in cold region particularly in the vicinity of the cold surface of sea or of a lake. Light rays starting from an objects (say a ship) are curved downward and on entering the eye the rays appear to come from S', thus giving an impression that the ship is floating in air. This effect is also called superior mirage.

An eye placed inside water sees the external world within a cone and rest surface of water appears as a vast sheet of mirror. Radius is

`r=h/sqrt(mu^2-1)`

Diamond and glass both shine if cut to the special shape, but diamond shine more than glass piece cut to same shape.

`text(Pole (vertex) ): ` Point of intersection of principal axis and the refracting surface.

`text(Focal Point (F) ) : ` It is an axial point having the property that any incident ray traveling parallel to the axis will after refraction, proceed toward, or appear to come from this point.

`text(Focal length (f) ) : ` The distance of the focus from the vertex of the refracting surface is called focal length. There is a great significance of the sign of focal length as it is able to state whether the spherical refracting surface is converging or diverging as in the chart.

`text(Sign of f : )`

+ ve `->` Converging nature of system.
- ve `->` Diverging nature of system.

In this section we describe how images are formed when light rays are refracted at the boundary between two transparent materials.

Consider two transparent media having indices of refraction `mu_1` and `mu_2`, where the boundary between the two media is a spherical surface of radius R (Fig.). We assume that the object at O is in the medium for which the index of refraction is `mu_1`. Let us consider the paraxial rays leaving O. As we shall see, all such rays are refracted at the spherical surface and focus at a single point `I`, the image point. Figure shows a single ray leaving point O and refracting to point I.

Snell's law of refraction applied to this ray gives

`mu_1sintheta_1=mu_2sintheta_2`

Because `theta_1` and `theta_2` are assumed to be small, we can use the small-angle approximation `sinthetaapproxtheta` (with angles in radians) and say that

`mu_1theta_1=mu_2theta_2`

Now we use the fact that an exterior angle of any triangle equals the sum of the two opposite interior angles. Applying this rule to triangles OPC and PIC in Figure gives

`theta_1=alpha+beta`
`beta=theta_2+gamma`

If we combine all three expressions and eliminate `theta_1`, and `theta_2`, we find that

`mu_1alpha+mu_2gamma=(mu_2-mu_1)beta`

From Figure, we see three right triangles that have a common vertical leg of length d. For paraxial rays (unlike the relatively large-angle ray shown in Fig.), the horizontal legs of these triangles are approximately p for the triangle containing angle `alpha`, R for the triangle containing angle `beta`, and q for the triangle containing angle `gamma` ). In the small-angle approximation, `tanalphaapproxalpha`, so we can write the approximate relationships from these triangles as follows:

`tanalphaapproxalpha=d/p`
`tanbetaapproxbeta=d/R`
`tangammaapproxgamma=d/q`

We substitute these expressions into above Equation and divide through by d to give

`(mu_2)/p + (mu_1)/q = (mu_2-mu_1)/R`

Let

u = object distance (with sign convention)
v = image distance (with sign convention)
R = Radius of curvature (with sign convention)

then
`p=-u`
`q=+v`
`R=+R`

putting the values of p, q and R in above equation we get

`(mu_2)/v -(mu_1)/u = (mu_2-mu_1)/R`

`text(Rules for image tracing for a linear, transverse extended object : )`

The basic rule is same as that in mirrors. Briefly,
(i) Draw a ray parallel to the principal axis which after refraction will be along the line passing through F.
(ii) Draw a ray incident along the line through centre it will pass undeviated.