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(D) The displacement method is based on the principle of reversibility of light. For a fixed distance $D$ between the object and the screen,if the len

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$\frac{80}{3}$

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(D) The displacement method is based on the principle of reversibility of light. For a fixed distance $D$ between the object and the screen,if the lens is at a distance $u$ from the object for the first position,it will be at a distance $v = D - u$ from the object for the second position.Given:Distance of first position from object,$u_1 = -40 \, cm$.Distance of second position from object,$u_2 = -80 \, cm$.Since the total distance $D$ between the object and the screen is constant,we have $D = 80 \, cm$.For the first position,the image distance $v_1 = D - |u_1| = 80 - 40 = 40 \, cm$.Using the lens formula,$\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$:$\frac{1}{f} = \frac{1}{40} - \frac{1}{-40} = \frac{1}{40} + \frac{1}{40} = \frac{2}{40} = \frac{1}{20}$.Wait,let's re-evaluate based on the standard displacement method formula: $f = \frac{D^2 - d^2}{4D}$,where $D$ is the distance between object and screen,and $d$ is the distance between the two lens positions.Here,$D = 80 \, cm$ and $d = 80 - 40 = 40 \, cm$.$f = \frac{80^2 - 40^2}{4 	imes 80} = \frac{6400 - 1600}{320} = \frac{4800}{320} = \frac{480}{32} = 15 \, cm$.Re-reading the question: The positions are $40 \, cm$ and $80 \, cm$ from the object. This implies $D = 80 \, cm$ and the two positions are $d_1 = 40 \, cm$ and $d_2 = 80 \, cm$ from the object. This is physically impossible as the screen must be at $80 \, cm$. If the screen is at $80 \, cm$,the second position cannot be at $80 \, cm$. Assuming the question implies $d = 80 - 40 = 40 \, cm$ and $D = 80 \, cm$,the calculation above holds. Given the options,the intended calculation is likely $\frac{1}{f} = \frac{1}{80} - \frac{1}{-40} = \frac{1+2}{80} = \frac{3}{80}$,so $f = \frac{80}{3} \, cm$.

In the displacement method,there are two positions of a lens for which we get a clear image on a screen. If the first position of the lens is at $40 \, cm$ from the object and the second position is at $80 \, cm$ from the object,the focal length of the lens is ......... $cm$.

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