To tackle this problem, we need to understand how a magnifying glass works, particularly in the context of a thin convex lens. We'll break this down into two parts: first, determining the closest and farthest distances for optimal viewing, and then calculating the angular magnification achievable with the lens.
Determining Distances for Optimal Reading
When using a magnifying glass, the goal is to create a virtual image that appears to be at the near point of the eye, which is typically around 25 cm for a person with normal vision. The thin convex lens forms a virtual image when the object is placed within its focal length. We can use the lens formula to find the required distances:
The Lens Formula
The lens formula is given by:
1/f = 1/v - 1/u
Where:
- f = focal length of the lens (5 cm)
- v = distance of the image from the lens
- u = distance of the object from the lens
For virtual images, the image distance (v) is considered negative when using the sign convention. In this case, we aim to have the virtual image appear at the near point, so v = -25 cm.
Calculating Object Distance (u)
Substituting into the lens formula:
1/5 = 1/(-25) - 1/u
This simplifies to:
1/u = 1/(-25) + 1/5
Finding a common denominator (which is 25) gives:
1/u = -1/25 + 5/25 = 4/25
Thus, we find:
u = 25/4 = 6.25 cm
This means the closest distance he should keep the lens from the page is approximately 6.25 cm.
Farthest Distance for Clear Viewing
To find the farthest distance, we can consider when the object is at the lens's focal length plus the near point. In this case, since we want the object to be at a distance greater than the focal length, we can place it at the focal point distance:
u = f (5 cm) is the maximum distance to achieve a virtual image at 25 cm. However, since the lens must be positioned within the focal length for proper magnification, the farthest distance for effective reading is slightly less than the focal length:
u_max = 25 cm (near point) + 5 cm (focal length) = 30 cm
Understanding Angular Magnification
Now, let’s discuss angular magnification or magnifying power. It describes how much larger an object appears when viewed through the lens compared to the naked eye.
Angular Magnification Formula
The formula for angular magnification (M) is:
M = (D/f) + 1
Where D is the near point distance (25 cm) and f is the focal length (5 cm).
Calculating Maximum and Minimum Angular Magnification
Substituting the values into the formula:
M = (25/5) + 1 = 5 + 1 = 6
This value represents the maximum angular magnification. Now, considering the minimum magnification occurs when the object is placed at the distance of the focal length:
M_min = (D/f) = (25/5) = 5
Summary of Findings
In summary:
- Closest distance for clear reading: 6.25 cm
- Farthest distance for effective viewing: just under 30 cm
- Maximum angular magnification: 6
- Minimum angular magnification: 5
These calculations illustrate how a simple convex lens can significantly enhance the ability to read small print, utilizing its optical properties effectively. Feel free to ask if there’s anything more you’d like to understand about this topic!
