91Ó°ÊÓ

To determine how far the lens needs to be from the camera sensor, we can use the thin lens formula. This fundamental formula connects three important variables: the object distance (\(d_o\)) which is how far the object is from the lens, the image distance (\(d_i\)) which is the distance from the lens to the sensor, and the focal length (\(f\)) of the lens. The formula is expressed as:
\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]By rearranging the terms, we isolate the image distance (\(d_i\)) to find out how far this needs to be from the lens:
\[\frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o}\]
Substituting the given focal length of 85 mm and the object distance of 3900 mm into this equation, we get:

\[\frac{1}{85} = \frac{1}{3900} + \frac{1}{d_i}\]

By calculating this, we solve for \(d_i\) and find that it is approximately 86.9 mm. Thus, the lens must be placed about 86.9 mm from the sensor for the image to be focused properly.

The magnification formula helps us understand how much larger or smaller the camera sensor's image is compared to the real-life object. It is defined by the ratio of image distance (\(d_i\)) over object distance (\(d_o\)):
\[M = \frac{d_i}{d_o}\]
In our exercise, this becomes:

\[M = \frac{86.9}{3900} \approx 0.0223\]

This value means the image on the sensor is significantly smaller than the original object because the magnification is less than 1. To figure out if the entire image fits on the sensor, you multiply magnification by the object's height:
\[h_i = M \times h_o\]

\[h_i = 0.0223 \times 1750 \approx 39.025 \, \text{mm}\]

Here, 39.025 mm represents the image height on the sensor. Since this exceeds the sensor's 36 mm longest side, the image does not fully fit.

Focal length is a key characteristic of lenses that determines how they focus light. It is the distance from the lens to the focal point, where parallel rays of light converge. For photography, this influences field of view and magnification power.
In this problem, the lens focal length is given as 85 mm. This value tells us the lens' ability to bring light into focus and plays a crucial role in both the image distance and magnification calculations.

• An 85 mm focal length is often used for portrait photography.
• It provides a balance of close-up detail and sufficient distance to avoid distortion.

By inputting the focal length into the thin lens formula, you can calculate how far the image will form from the lens, revealing the relationship between an object’s positioning and how it appears on a camera sensor.