The focal lengths of the objective lens and the eyepiece of a microscope are 0.50 cm and 2.0 cm, respectively, and their separation adjusted for minimum eyestrain is 6.0 cm. If the microscope is
Focused on a small object, what is the distance between the object and the objective lens?
A) 0.57 cm
B) 0.31 cm
C) 0.49 cm
D) 0.73 cm
E) 0.42 cm

Question

Quizplus · Accepted Answer

The total magnification of a microscope is the product of the magnifications of the objective lens and the eyepiece. The magnification of the objective lens, $M_o$, is given by the equation $M_o = \frac{d_i}{f_o}$, where $d_i$ is the image distance for the objective lens, and $f_o$ is the focal length of the objective lens. The image formed by the objective lens acts as the object for the eyepiece, and for minimum eyestrain, the final image is formed at infinity. This means the object distance for the eyepiece, $d_o'$, equals its focal length, $f_e$, because when an object is at the focal point of a lens, the image is formed at infinity. The total length of the microscope, $L$, is the sum of the image distance of the objective lens and the object distance of the eyepiece, so $L = d_i + f_e$.Given:- $f_o = 0.50$ cm (focal length of the objective lens)- $f_e = 2.0$ cm (focal length of the eyepiece)- $L = 6.0$ cm (total length)We need to find the object distance for the objective lens, $d_o$, which is not directly given but can be inferred. Since $L = d_i + f_e$ and we know $L$ and $f_e$, we can solve for $d_i$. However, to find $d_o$, we use the lens equation $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$ for the objective lens.First, find $d_i$ using $L = d_i + f_e$:$6.0 = d_i + 2.0$$d_i = 4.0$ cmThen, use the lens equation for the objective lens to find $d_o$:$\frac{1}{0.50} = \frac{1}{d_o} + \frac{1}{4.0}$Solving for $d_o$ gives:$\frac{1}{d_o} = \frac{1}{0.50} - \frac{1}{4.0}$$\frac{1}{d_o} = 2 - 0.25$$\frac{1}{d_o} = 1.75$$d_o = \frac{1}{1.75}$$d_o \approx 0.57$ cmTherefore, the distance between the object and the objective lens is approximately 0.57 cm.

The Focal Lengths of the Objective Lens and the Eyepiece