To solve the problem of how much the eyepiece of the microscope must be moved to refocus the image after removing a lens from the objective, we need to understand a few key concepts about lenses and magnification in a compound microscope. Let's break this down step by step.
Understanding the Microscope Setup
A compound microscope typically consists of two main lenses: the objective lens and the eyepiece. The objective lens is responsible for forming a real image of the object, while the eyepiece magnifies this image for the viewer's eye. The focal length of the objective lens is crucial for determining how the image is formed.
Initial Conditions
- Distance of the object from the objective (u) = -0.03 m (the negative sign indicates that the object is on the same side as the incoming light).
- Focal length of the objective lens (f) = 0.02 m.
Finding the Image Distance (v) for the Objective Lens
We can use the lens formula, which is given by:
1/f = 1/v - 1/u
Substituting the known values:
1/0.02 = 1/v - 1/(-0.03)
This simplifies to:
1/0.02 = 1/v + 1/0.03
To solve for v, we first find a common denominator:
1/v = 1/0.02 - 1/0.03
Calculating the right side:
1/v = (3 - 2)/0.06 = 1/0.06
Thus, we find:
v = 0.06 m
Effect of Removing a Lens
Now, when a lens with a focal length of 0.1 m is removed from the objective, the effective focal length of the objective lens changes. We need to find the new effective focal length (f') of the remaining lenses. For two lenses in contact, the formula for the combined focal length is:
1/f' = 1/f1 + 1/f2
Here, f1 = 0.02 m and f2 = 0.1 m. Thus:
1/f' = 1/0.02 + 1/0.1
Calculating this gives:
1/f' = 50 + 10 = 60
So, the new focal length is:
f' = 1/60 m ≈ 0.01667 m
Finding the New Image Distance (v') for the New Objective Lens
Now we apply the lens formula again with the new focal length:
1/f' = 1/v' - 1/u
Substituting the values:
1/0.01667 = 1/v' + 1/0.03
Rearranging gives:
1/v' = 60 - 33.33 = 26.67
Thus, we find:
v' = 1/26.67 m ≈ 0.0375 m
Calculating the Movement of the Eyepiece
The distance the eyepiece must be moved can be determined by the difference in image distances:
Distance moved = v' - v
Substituting the values we found:
Distance moved = 0.0375 m - 0.06 m = -0.0225 m
This negative value indicates that the eyepiece must be moved closer to the objective lens by approximately 0.0225 m, or 22.5 mm, to refocus the image.
Final Thoughts
In summary, when a lens is removed from the objective of a compound microscope, the effective focal length changes, which in turn affects the image distance. By recalculating the image distance with the new focal length, we can determine how much the eyepiece needs to be adjusted to maintain a clear focus on the object.
