Name: Total Internal Reflection From Prism
Uploaded: 2019-07-19
Duration: 1 min 56 s
Description: You must have seen reflectors on a bicycle. Why are they attached to the mudguards? What use do they have? Let’s watch this video on reflectors to find out.

Question

A convex lens of focal length

20 cm

and a concave lens of focal length

10 cm

are placed

10 cm

apart with their principal axes coinciding. A beam of light travelling parallel to the principal axis and having a beam of diameter

5.0 mm

, is incident on the combination. Show that the emergent beam is parallel to the incident one. Find the beam diameter of the emergent beam.

Accepted Answer

Given,

The focal length of convex lens, $F_{1} = + 20 cm$ .

The focal length of concave lens, $F_{2} = - 10 cm$ .

The distance between the two lens $= 10 cm$ .

If the ray is incident from the convex side,

$I_{1} =$ the image formed by the converging lens.

Formation of $I_{1}$ ,

Parallel rays to the principal axis will converge to principle focus.

$I_{1}$ will act as an object to the diverging lens.

Formation of final image,

Lens equation,

$\frac{1}{v_{1}} - \frac{1}{u_{1}} = \frac{1}{f_{2}}$

where, $v_{1} =$ image distance from concave lens

$u_{1} =$ object distance from lens $= + 10 cm$ .

$\frac{1}{v_{1}} - \frac{1}{10} = \frac{1}{- 10}$

$v_{1} \to \infty$ .

Rays become parallel to principle axis.

$Δ A B I_{1} ~ Δ C D I_{1}$

(Since, corresponding angles are equal.)

$\frac{A B}{C D} = \frac{B I_{1}}{D I_{1}}$

$\frac{\frac{5}{2} mm}{C D} = \frac{20 cm}{10 cm}$

$C D = 1 .25 mm$ .

$∴$ Diameter of emergent beam $= 2 \times 1 .25 mm = 2 .5 mm$ .

If ray is incident from concave lens side,

$I_{2} =$ image formed by diverging lens.

Formation of $I_{2}$ ,

Parallel rays to principal axis will appear to diverge act from principle focus.

$I_{2}$ will act as an object to converging lens.

Formation of final image,

Lens equation,

$\frac{1}{v_{2}} - \frac{1}{u_{2}} = \frac{1}{f_{1}}$

where, $v_{2} =$ image distance from concave lens

$u_{2} =$ object distance from lens $= - 20 cm$ .

$\frac{1}{v_{2}} - \frac{1}{- 20} = \frac{1}{+ 20}$

$v_{2} \to \infty$ .

Rays become parallel to principle axis.

$Δ A B I_{1} ~ Δ C D I_{2}$

(Since, corresponding angles are equal.)

$\frac{A B}{C D} = \frac{B I_{1}}{D I_{1}}$

$\frac{A B}{\frac{5}{2} mm} = \frac{20 cm}{10 cm}$

$C D = 5 mm$ .

$∴$ Diameter of emergent beam $= 2 \times 5 mm = 10 mm = 1 cm$ .

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