For a microscope on which 17 ocular units align with 100 μ m on the stage micrometer when the 4x objective is used, how many micrometers are there per ocular unit when using the (a) 4x objective, (b) 10x objective, (c) 45x objective, and (d) 97x objective?
The correct answer and explanation is :
Let’s work through the problem carefully:
Given:
- 17 ocular units align with 100 μm on the stage micrometer when using the 4x objective.
First, find the size (in micrometers) of one ocular unit at 4x:
[
\text{Size per ocular unit at 4x} = \frac{100\ \mu m}{17} \approx 5.88\ \mu m
]
Now, when you change the objective lens (to 10x, 45x, or 97x), the number of micrometers per ocular unit decreases proportionally because magnification increases.
Since magnification and field size are inversely proportional, we can calculate the new micrometers per ocular unit by adjusting for the ratio of magnifications:
The formula:
[
\text{New micrometers per ocular unit} = \text{(Micrometers per ocular unit at 4x)} \times \frac{4}{\text{New objective magnification}}
]
(a) 4x Objective
Already calculated:
[
5.88\ \mu m\ \text{per ocular unit}
]
(b) 10x Objective
[
5.88\ \mu m \times \frac{4}{10} = 2.35\ \mu m\ \text{per ocular unit}
]
(c) 45x Objective
[
5.88\ \mu m \times \frac{4}{45} \approx 0.523\ \mu m\ \text{per ocular unit}
]
(d) 97x Objective
[
5.88\ \mu m \times \frac{4}{97} \approx 0.242\ \mu m\ \text{per ocular unit}
]
Final Answers:
- (a) 5.88 μm per ocular unit (4x objective)
- (b) 2.35 μm per ocular unit (10x objective)
- (c) 0.523 μm per ocular unit (45x objective)
- (d) 0.242 μm per ocular unit (97x objective)
Explanation (300+ words)
In microscopy, calibration is essential to accurately measure the size of specimens. Calibration typically uses a stage micrometer, a microscope slide with a precisely ruled scale (usually in micrometers), and the ocular micrometer, which is a glass disk with an arbitrary scale inserted into the microscope eyepiece.
The relationship between the ocular micrometer units and actual distances must be established for each objective lens because magnification changes the apparent size of features. The higher the magnification, the smaller the actual size represented by each ocular division.
In this problem, it is given that with the 4x objective, 17 ocular units line up with 100 μm on the stage micrometer. Dividing 100 μm by 17 gives approximately 5.88 μm per ocular unit at 4x magnification.
When you switch objectives, the apparent size of an object increases (higher magnification), so each ocular division represents a smaller real distance. Since magnification and measured size are inversely proportional, we adjust by multiplying the original calibration (5.88 μm) by the ratio of the original magnification (4x) to the new magnification.
Thus, when switching to 10x, the value per ocular unit decreases to 2.35 μm. At 45x, it further decreases to about 0.523 μm, and at 97x, it becomes about 0.242 μm per ocular unit.
This method ensures accurate measurement at any magnification without needing to recalibrate each time physically. However, for highly precise work (like research microscopy), calibration is often repeated at each magnification to account for slight mechanical imperfections.