Question

The length of the compound microscope is 14cm. The magnifying power for a relaxed eye is 25. If the focal length of the eye lens is 5cm, then the object distance for the objective lens will be:
$\begin{aligned} & (A)1.8cm \\\ & (B)1.5cm \\\ & (C)2.1cm \\\ & (D)2.4cm \\\ \end{aligned}$

Answer 1

Since, it has been given that the eyes are relaxed it means the final image after refraction from the eye lens forms at infinity. This means the image formed by the objective lies at the focus of the eye lens. That is, the length of the microscope is equal to the sum of focal length of the eye lens and the distance at which the object is making the image.

Complete answer:
Let the distance of image from the pole of the objective lens be denoted by ${{v}_{o}}$ . Also, let the focus of the eye lens be denoted by ${{f}_{e}}$ . Then, we can write the length of the compound microscope (L) as:
$\Rightarrow L={{v}_{o}}+{{f}_{e}}$
Here,

& \Rightarrow L=14cm \\\ & \Rightarrow {{f}_{e}}=5cm \\\ \end{aligned}$$ Putting these values in the above expression, we get: $\begin{aligned} & \Rightarrow {{v}_{o}}=L-{{f}_{e}} \\\ & \Rightarrow {{v}_{o}}=14-5cm \\\ & \Rightarrow {{v}_{o}}=9cm \\\ \end{aligned}$ Now, let the distance at which the object is for the objective lens be ${{u}_{o}}$ . Then, using the formula of magnifying power of microscope for relaxed eyes, we get: $\Rightarrow M=\dfrac{{{v}_{o}}}{{{u}_{o}}}\times \dfrac{D}{{{f}_{e}}}$ Here, we have: $\Rightarrow M=25$ $\begin{aligned} & \Rightarrow {{v}_{o}}=9cm \\\ & \Rightarrow D=25cm \\\ & \Rightarrow {{f}_{e}}=5cm \\\ \end{aligned}$ Putting all these values in the above equation and solving for the object distance, we get: $\begin{aligned} & \Rightarrow {{u}_{o}}=\dfrac{{{v}_{o}}}{M}\times \dfrac{D}{{{f}_{e}}} \\\ & \Rightarrow {{u}_{o}}=\dfrac{9}{25}\times \dfrac{25}{5}cm \\\ & \Rightarrow {{u}_{o}}=\dfrac{9}{5}cm \\\ & \therefore {{u}_{o}}=1.8cm \\\ \end{aligned}$ Hence, the object distance for the objective lens of the compound microscope will be 1.8cm **Hence, option (A) is the correct option.** **Note:** In this problem we applied the concepts of refraction and formulas of microscope together. Problems like these should be considered at a hard level and one should practice such problems more and more as it gives us a different spectrum of analogies while solving a complex question in exams. Also, one should at least remember all the formulas and conditions so that if a similar problem appears again, it will be faster to calculate.