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JEE Mains

## Limit of resolution of optical instruments

In determining the limit of resolution of optical instruments like a telescope or a microscope, for the two stars to be just resolved,

implying
Thus will be small if the diameter of the objective is large. This implies that the telescope will have better resolving power if is large. It is for this reason that for better resolution, a telescope must have a large diameter objective.

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## Image formation for lenses with one side silvered

A plano convex lens of focal length has its plane surface silvered. An object is placed from the lens on the convex side. The distance of the image from the lens is:

or,
or,
or,
Now, this image acts as the object for the lens.
So,

or,
or,
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## Angle of deviation  Angle of deviation () is the angle between emergent ray and incident ray.
For a single refracting surface,
For a prism,
where is the angle of the prism.
For angle of minimum deviation, is minimum and

For small ,
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## combination of lenses  Problem:
A convex lens, of focal length , a concave lens of focal length , and a plane mirror are arranged as shown. For an object kept at a distance of from the convex lens, the final image, formed by the combination, is a real image, at a distance of :
Solution:
Location of image formation after first refraction from the lens,.
This gives
Next refraction from concave lens,
Hence
Since this will be formed behind the mirror, this will be final image.It is at a distance of from the concave lens.
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## Apparent depth and real depth  Real Depth is actual distance of an object beneath the surface, as would be measured by submerging a perfect ruler along with it.

Apparent depth in a medium is the depth of an object in a denser medium as seen from the rarer medium. Its value is smaller than the real depth.
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## Transmission of light from a denser medium to a rarer medium at different angles of incidence  When a ray of light enters from a denser medium to a rarer medium,it bends away from the normal, for example, the ray .The incident ray is partially reflected () and partially transmitted() or refracted, the angle of refraction (r ) being larger than the angle of incidence (i ). As the angle of incidence increases, so does the angle of refraction, till for the ray , the angle of refraction is .The refracted ray is bent so much away from the normal that it grazes the surface at the interface between the two media. This is shown by the ray If the angle of incidence is increased still further (e.g.,the ray ), refraction is not possible, and the incident ray is totally reflected.
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## Problems on Total internal reflection in Prism  Example: At what angle should a ray of light be incident on the face of a prism of refracting angle so that it just suffers total internal reflection at the other face. Find the refractive index of the prism.Refractive index of prism is 1.524.

Solution:

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## Speed of object and speed of image

Suppose while sitting in a parked car, you notice a jogger approaching towards you in the side view mirror of . If the jogger is running at a speed of , how fast the image of the jogger appear to move when the jogger is away?
Solution:
From the mirror equation, we get:

For convex mirror, since , then for ,
Since the jogger moves at a constant speed of , after , the position of the image (for ) is .
The shift in the position of image in is

Therefore, the average speed of the image when the jogger is between and from the mirror, is .
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## Refraction of rays at spherical surfaces  10

## Lens Maker's Formula  Consider a convex lens (or concave lens) of absolute refractive index  to be placed in a rarer medium of absolute refractive index . Considering the refraction of a point object on the surface , the image is formed at  who is at a distance of .

(as the lens is thin)

It follows from the refraction due to convex spherical surface

The refracted ray from A suffers a second refraction on the surface and emerges along BI. Therefore I is the final real image of O.

Here the object distance is

is very small

Let

(Final image distance)

Let  be radius of curvature of second surface of the lens. It follows from refraction due to concave spherical surface from denser to rarer medium that

or

But and

Thus we get=

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## Combination of thin lenses in contact  Consider two lenses A and B of focal length and placed in contact with each other.
An object is placed at O on  the common principal axis. The lens A produces an image at and this image acts as the object for the second lens B. The final image is produced at as shown in figure.
PO = u, object distance for the first lens (A),
PI = v, final image distance and
, image distance for the first lens (A) and also object distance for second lens (B).
For the image produced by the first lens A,
.... (1)
For the final image I, produced by the second lens B,
... (2)
... (3)
If the combination is replaced by a single lens of focal length F such that it forms the image of O at the same position I, then
... (4)
From equations (3) and (4),
...... (5)
This F is the focal length of the equivalent lens for the combination.
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## combination of lenses  Problem:
A convex lens, of focal length , a concave lens of focal length , and a plane mirror are arranged as shown. For an object kept at a distance of from the convex lens, the final image, formed by the combination, is a real image, at a distance of :
Solution:
Location of image formation after first refraction from the lens,.
This gives
Next refraction from concave lens,
Hence
Since this will be formed behind the mirror, this will be final image.It is at a distance of from the concave lens.