Problem 47 - University Physics with Modern Physics

Question

When a camera is focused, the lens is moved away from or toward the digital image sensor. If you take a picture of your friend, who is standing 3.90 m from the lens, using a camera with a lens with an 85-mm focal length, how far from the sensor is the lens? Will the whole image of your friend, who is 175 cm tall, fit on a sensor that is 24 mm \(\times\) 36 mm?

Step-by-Step Solution

Answer

The lens is 86.9 mm from the sensor, and the whole image will not fit on the sensor.

Step 1: Understand the Lens Formula

The formula that relates object distance \(d_o\), image distance \(d_i\), and focal length \(f\) for a thin lens is: \[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]Here, \(f = 85\) mm is the focal length and the object distance \(d_o = 3.90\) m or 3900 mm from the lens. We need to find \(d_i\), the distance from the lens to the sensor.

Step 2: Substitute Values into the Lens Formula

Substitute the known values into the lens equation:\[ \frac{1}{85} = \frac{1}{3900} + \frac{1}{d_i} \]Rearrange to solve for \( \frac{1}{d_i} \):\[ \frac{1}{d_i} = \frac{1}{85} - \frac{1}{3900} \]

Step 3: Calculate Image Distance

Calculate \( \frac{1}{d_i} \) using the values:\[ \frac{1}{d_i} = \frac{1}{85} - \frac{1}{3900} \approx 0.0117647 - 0.00025641 \approx 0.0115083 \]Therefore, \(d_i = \frac{1}{0.0115083} \approx 86.9\) mm.

Step 4: Check if Image Fits on the Sensor

Use the magnification formula \( M = \frac{d_i}{d_o} \) to find the size of the image. Calculate the magnification:\[ M = \frac{86.9}{3900} \approx 0.0223 \]The height of the image \( h_i \) is given by \( h_i = M \times h_o \), where \( h_o = 1750 \) mm:\[ h_i = 0.0223 \times 1750 \approx 39.025 \text{ mm} \]Check if this fits within the sensor's longest dimension, which is 36 mm. 39.025 mm is larger than 36 mm, so the whole image of the friend does not fit on the sensor.

Key Concepts

Image Distance Calculation

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.

Magnification Formula

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

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.