Advertisements
Advertisements
Question
In many experimental set-ups the source and screen are fixed at a distance say D and the lens is movable. Show that there are two positions for the lens for which an image is formed on the screen. Find the distance between these points and the ratio of the image sizes for these two points.
Advertisements
Solution
From the reversibility of u and v, as seen from the formula for lens,
`1/f = 1/v - 1/u`
It is clear that there are two positions for which there shall be an image on the screen.
Let the first position be when the lens is at O.
Given `-u + v = D`
⇒ `u = -(D - v)`
Placing it in the lens formula
`1/(D - v) + 1/v = 1/f`
⇒ `(v + D - v)/((D - v)v) = 1/f`
⇒ `v^2 - Dv + Df` = 0
⇒ `v^2 = D/2 +- sqrt(D^2 - 4Df)/2`
`u = -(D - v) = - (D/2 + sqrt(D^2 - 4Df)/2)`
Thus, if the object distance is `D/2 - sqrt(D^2 - 4Df)/2` then the image is at `D/2 + sqrt(D^2 - 4Df)/2`
If the object distance is `D/2 + sqrt(D^2 - 4Df)/2`, then the image is at `D/2 - sqrt(D^2 - 4Df)/2`.
The distance between the poles for these two object distances is
`D/2 - sqrt(D^2 - 4Df)/2 - (D/2 - sqrt(D^2 - 4Df)/2) = sqrt(D^2 - 4Df)`
Let `d = sqrt(D^2 - 4Df)`
If `u = D/2 + d/2` then the image is at `v = D/2 - d/2`
∴ The magnification `m_1 = (D - d)/(D + d)`
If `u = (D - d)/2` then `v = (D + d)/2`
∴ The magnification `m_2 = (D + d)/(D - d)` Thus `m_2/m_1 = ((D + d)/(D - d))^2`.
APPEARS IN
RELATED QUESTIONS
A beam of light converges at a point P. Now a lens is placed in the path of the convergent beam 12 cm from P. At what point does the beam converge if the lens is
- a convex lens of focal length 20 cm, and
- a concave lens of focal length 16 cm?
An object of size 3.0 cm is placed 14 cm in front of a concave lens of focal length 21 cm. Describe the image produced by the lens. What happens if the object is moved further away from the lens?
You have learnt that plane and convex mirrors produce virtual images of objects. Can they produce real images under some circumstances? Explain.
The image of a small electric bulb fixed on the wall of a room is to be obtained on the opposite wall 3 m away by means of a large convex lens. What is the maximum possible focal length of the lens required for the purpose?
A screen is placed 90 cm from an object. The image of the object on the screen is formed by a convex lens at two different locations separated by 20 cm. Determine the focal length of the lens.
An object 1.5 cm in size is placed on the side of the convex lens in the arrangement (a) above. The distance between the object and the convex lens is 40 cm. Determine the magnification produced by the two-lens system, and the size of the image
A man with normal near point (25 cm) reads a book with small print using a magnifying glass: a thin convex lens of focal length 5 cm.
(a) What is the closest and the farthest distance at which he should keep the lens from the page so that he can read the book when viewing through the magnifying glass?
(b) What is the maximum and the minimum angular magnification (magnifying power) possible using the above simple microscope?
A card sheet divided into squares each of size 1 mm2 is being viewed at a distance of 9 cm through a magnifying glass (a converging lens of focal length 9 cm) held close to the eye.
- What is the magnification produced by the lens? How much is the area of each square in the virtual image?
- What is the angular magnification (magnifying power) of the lens?
- Is the magnification in (a) equal to the magnifying power in (b)? Explain.
A convex lens forms a real image of a point object placed on its principals axis. If the upper half of the lens is painted black,
(a) the image will be shifted downward
(b) the image will be shifted upward
(c) the image will not be shifted
(d) the intensity of the image will decrease.
A small piece of wood is floating on the surface of a 2.5 m deep lake. Where does the shadow form on the bottom when the sum is just setting? Refractive index of water = 4/3.
Answer the following question.
An optical instrument uses a lens of 100 D for the objective lens and 50 D for its eyepiece. When the tube length is kept at 20 cm, the final image is formed at infinity.
(a) Identify the optical instrument.
(b) Calculate the magnification produced by the instrument.
An object approaches a convergent lens from the left of the lens with a uniform speed 5 m/s and stops at the focus. The image ______.
Will the focal length of a lens for red light be more, same or less than that for blue light?
An unsymmetrical double convex thin lens forms the image of a point object on its axis. Will the position of the image change if the lens is reversed?
In the given figure the radius of curvature of the curved face in the planoconvex and the planoconcave lens is 15 cm each. The refractive index of the material of the lenses is 1.5. Find the final position of the image formed.
