A microscope has an objective lens of focal length 1.40 mm and an eyepiece of focal length 20.00 mm. It is adjusted for minimum eyestrain for persons with a near point of 25.0 cm. A blood
Sample is placed 1.50 mm from the objective. How far apart are the lenses?
A) 23 mm
B) 21 mm
C) 41 mm
D) 20 mm

Question

Quizplus · Accepted Answer

For minimum eyestrain, the image formed by the objective lens is at the near point of the observer, which is 25.0 cm (250 mm) from the eyepiece. The magnification of the objective lens, $M_o$, is given by the ratio of the image distance ($v_o$) to the object distance ($u_o$), where $u_o$ is negative because it is on the opposite side of the incoming light. The lens formula is $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$, where $f$ is the focal length, $v$ is the image distance, and $u$ is the object distance.For the objective lens:$$ \frac{1}{f_o} = \frac{1}{v_o} - \frac{1}{u_o} $$Given $f_o = 1.40$ mm and $u_o = -1.50$ mm (negative because the object is on the opposite side of the direction of incoming light), we can solve for $v_o$, the image distance from the objective lens, which will also be the object distance for the eyepiece since the image formed by the objective acts as the object for the eyepiece.$$ \frac{1}{1.40} = \frac{1}{v_o} + \frac{1}{1.50} $$$$ \frac{1}{v_o} = \frac{1}{1.40} - \frac{1}{1.50} $$$$ v_o = \frac{1}{(\frac{1}{1.40} - \frac{1}{1.50})} \approx 42.00 \text{ mm} $$This is the distance from the objective lens to its image. Since this image acts as the object for the eyepiece and is to be placed at the near point (250 mm from the eyepiece), the total distance between the objective and eyepiece lenses is the sum of $v_o$ and the distance to the near point from the eyepiece.However, the calculation above directly gives us the distance from the objective lens to the point where the image is formed to be viewed through the eyepiece. Since the question asks for the distance between the lenses, and considering the image formed by the objective lens needs to be at the focal point of the eyepiece for it to be viewed at infinity (for minimum eyestrain), we adjust our approach:Given the image is formed at the near point (250 mm from the eyepiece), and considering the lens formula for the eyepiece, we need to find the distance between the lenses when the final image is at the near point.The correct approach involves understanding that the image distance from the objective lens ($v_o$) plus the object distance for the eyepiece ($u_e$) (which is negative in this calculation) equals the distance between the lenses. Since the final image is at the observer's near point, the calculation simplifies to finding how the system achieves this configuration.The distance between the lenses is essentially the length of the tube, which is the sum of the focal length of the objective and the focal length of the eyepiece plus any additional distance needed to ensure the image is at the near point. The calculation provided was incorrect in its approach to solving the problem, as it did not correctly apply the principles of optical imaging in compound microscopes.The correct calculation involves using the optical bench equation and considering the final image position relative to the eyepiece. The distance between the lenses (tube length) in a microscope adjusted for minimum eyestrain (where the final image is at infinity) is not directly derived from the given information without considering the specific configuration of the microscope for viewing a distant image comfortably.Given the error in the initial explanation, the correct approach to find the distance between the lenses involves considering both the object and image distances relative to each lens and ensuring the final image is at a comfortable viewing position (at infinity for minimum eyestrain), which typically involves the eyepiece focusing on an image formed by the objective lens just outside its focal point. The calculation should account for the optical properties of both lenses and their arrangement to achieve the desired viewing condition.

A Microscope Has an Objective Lens of Focal Length 1