Advertisements
Advertisements
प्रश्न
A short object of length L is placed along the principal axis of a concave mirror away from focus. The object distance is u. If the mirror has a focal length f, what will be the length of the image? You may take L << |v – f|.
Advertisements
उत्तर
As the mean distance of object from mirror is u
∴ `u_1 = u - L/2` and `u_2 = (u + L/2)`
Let the image of the two ends of object form at distance v1 and v2(v1 > v2). So length of image on principal axis is L' = (v1 – v2)
`1/v + 1/u = 1/f` or `1/v + 1/f = 1/u`
⇒ `1/v = (u - f)/(uf)`
⇒ `v = (uf)/(u - f)`
So L' = `v_1 - v_2 = ((u - L/2)f)/((u - L/2) - f) - ((u + L/2)f)/((u + L/2) - f)`
⇒ L' = `f (u - L/2)/((u - f - L/2)) - (u + 1/2)/((u - f + L/2))`
⇒ L' = `f[((u - L/2)(u - f + L/2) - (u + L/2)(u - f - L/2))/((u - f - L/2)(u - f + 1/2))]`
= `(f[u^2 - uf + (uL)/2 - (uL)/2 + (fL)/2 - L^2/4 - (u^2 - uf - (uL)/2 + (uL)/2 - (fL)/2 - L^2/4)])/((u - f)^2 - L^2/4)`
∴ `L < < (u - f)`
∴ `L^2/4 < < < (u - f)`
So neglecting the terms `L^2/4`
`L^2 = f[((fL)/2 + (fL)/2)/(u - f)^2]`
`L^2 = (ffL)/((u - f)^2` ⇒ `L^2 = (Lf^2)/((u - f)^2`
It is the length of image f.
APPEARS IN
संबंधित प्रश्न
A small candle, 2.5 cm in size is placed at 27 cm in front of a concave mirror of radius of curvature 36 cm. At what distance from the mirror should a screen be placed in order to obtain a sharp image? Describe the nature and size of the image. If the candle is moved closer to the mirror, how would the screen have to be moved?
A 4.5 cm needle is placed 12 cm away from a convex mirror of focal length 15 cm. Give the location of the image and the magnification. Describe what happens as the needle is moved farther from the mirror.
A double convex lens is made of a glass of refractive index 1.55, with both faces of the same radius of curvature. Find the radius of curvature required, if the focal length is 20 cm.
Can mirrors give rise to chromatic aberration?
Following figure shows three transparent media of refractive indices \[\mu_1 , \mu_2 \text{ and } \mu_3\]. A point object O is placed in the medium \[\mu_2\]. If the entire medium on the right of the spherical surface has refractive index \[\mu_3\], the image forms at O". In the situation shown,
A spherical surface of radius 30 cm separates two transparent media A and B with refractive indices 1.33 and 1.48 respectively. The medium A is on the convex side of the surface. Where should a point object be placed in medium A so that the paraxial rays become parallel after refraction at the surface?
A narrow pencil of parallel light is incident normally on a solid transparent sphere of radius r. What should be the refractive index is the pencil is to be focussed (a) at the surface of the sphere, (b) at the centre of the sphere.
A converging lens of focal length 12 cm and a diverging mirror of focal length 7.5 cm are placed 5.0 cm apart with their principal axes coinciding. Where should an object be placed so that its image falls on itself?
A converging lens and a diverging mirror are placed at a separation of 15 cm. The focal length of the lens is 25 cm and that of the mirror is 40 cm. Where should a point source be placed between the lens and the mirror so that the light, after getting reflected by the mirror and then getting transmitted by the lens, comes out parallel to the principal axis?
A converging lens of focal length 40 cm is kept in contact with a diverging lens of focal length 30 cm. Find the focal length of the combination .
How can the spherical aberration produced by a lens be minimized?
With the help of a ray diagram, obtain the relation between its focal length and radius of curvature.
Focal length of a mirror is given by ______.
A thin converging lens of focal length 12 cm is kept in contact with a thin diverging lens of focal length 18 cm. Calculate the effective/equivalent focal length of the combination.
An astronomical refractive telescope has an objective of focal length 20 m and an eyepiece of focal length 2 cm.
- The length of the telescope tube is 20.02 m.
- The magnification is 1000.
- The image formed is inverted.
- An objective of a larger aperture will increase the brightness and reduce chromatic aberration of the image.
(i) Consider a thin lens placed between a source (S) and an observer (O) (Figure). Let the thickness of the lens vary as `w(b) = w_0 - b^2/α`, where b is the verticle distance from the pole. `w_0` is a constant. Using Fermat’s principle i.e. the time of transit for a ray between the source and observer is an extremum, find the condition that all paraxial rays starting from the source will converge at a point O on the axis. Find the focal length.
(ii) A gravitational lens may be assumed to have a varying width of the form
`w(b) = k_1ln(k_2/b) b_("min") < b < b_("max")`
= `k_1ln (K_2/b_("min")) b < b_("min")`
Show that an observer will see an image of a point object as a ring about the center of the lens with an angular radius
`β = sqrt((n - 1)k_1 u/v)/(u + v)`
Parallel rays striking a spherical mirror far from the optic axis are focussed at a different point than are rays near the axis thereby the focus moves toward the mirror as the parallel rays move toward the outer edge of the mirror. What value of incidence angle θ produces a 2% change in the location of the focus, compared to the location for θ very close to zero?
A concave mirror of focal length 12 cm forms three times the magnified virtual image of an object. Find the distance of the object from the mirror.
If an object is placed at a distance of 10 cm in front of a concave mirror of a focal length of 20 cm, the image formed will be ______.
A lens of focal length f is divided into two equal parts and then these parts are put in a combination as shown in the figure below.
- What is the focal length of L1?
- What is the focal length of the final combination?
