Question

A luminous object and a screen are at a fixed distance $$D$$ apart. A converging lens of focal length $$f$$ is placed between the object and screen. A real image of the object in formed on the screen for two lens positions of they are separated by a distance $$d$$ equal to

A. $$\sqrt {D\left( {D + 4f} \right)} $$
B. $$\sqrt {D\left( {D - 4f} \right)} $$  
C. $$\sqrt {2D\left( {D - 4f} \right)} $$
D. $$\sqrt {{D^2} + 4f} $$
Answer :   $$\sqrt {D\left( {D - 4f} \right)} $$
Solution :
Let the object distance be $$x.$$ Then, the image distance is $$D - x.$$
From lens equation, $$\frac{1}{x} + \frac{1}{{D - x}} = \frac{1}{f}$$
On algebraic rearrangement, we get
$$x2 - Dx + Df = 0$$
On solving for $$x,$$ we get
$$\eqalign{ & {x_1} = \frac{{D - \sqrt {D\left( {D - 4f} \right)} }}{2} \cr & {x_2} = \frac{{D + \sqrt {D\left( {D - 4f} \right)} }}{2} \cr} $$
The distance between the two object positions is
$$d = {x_2} - {x_1} = \sqrt {D\left( {D - 4f} \right)} $$

Releted MCQ Question on
Optics and Wave >> Ray Optics

Releted Question 1

When a ray of light enters a glass slab from air,

A. Its wavelength decreases.
B. Its wavelength increases.
C. Its frequency decreases.
D. neither its wavelength nor its frequency changes.
Releted Question 2

A glass prism of refractive index 1.5 is immersed in water (refractive index $$\frac{4}{3}$$ ). A light beam incident normally on the face $$AB$$  is totally reflected to reach on the face $$BC$$  if
Ray Optics mcq question image

A. $$\sin \theta \geqslant \frac{8}{9}$$
B. $$\frac{2}{3} < \sin \theta < \frac{8}{9}$$
C. $$\sin \theta \leqslant \frac{2}{3}$$
D. None of these
Releted Question 3

A ray of light from a denser medium strike a rarer medium at an angle of incidence $$i$$ (see Fig). The reflected and retracted rays make an angle of $${{{90}^ \circ }}$$ with each other. The angles of reflection and refraction are $$r$$ and $$r’$$ The critical angle is
Ray Optics mcq question image

A. $${\sin ^{ - 1}}\left( {\tan r} \right)$$
B. $${\sin ^{ - 1}}\left( {\tan i} \right)$$
C. $${\sin ^{ - 1}}\left( {\tan r'} \right)$$
D. $${\tan ^{ - 1}}\left( {\sin i} \right)$$
Releted Question 4

A ray of light from a denser medium strike a rarer medium at an angle of incidence $$i$$ (see Fig). The reflected and retracted rays make an angle of $${{{90}^ \circ }}$$ with each other. The angles of reflection and refraction are $$r$$ and $$r’$$ The critical angle is
Ray Optics mcq question image

A. $${\sin ^{ - 1}}\left( {\tan r} \right)$$
B. $${\sin ^{ - 1}}\left( {\tan i} \right)$$
C. $${\sin ^{ - 1}}\left( {\tan r'} \right)$$
D. $${\tan ^{ - 1}}\left( {\sin i} \right)$$

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