Problem 5 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A 105 -mm-focal-length lens is used to focus an image on the sensor of a camera. The maximum distance allowed between the lens and the sensor plane is \(132 \mathrm{~mm} .\) (a) How far ahead of the sensor should the lens be if the object to be photographed is \(10.0 \mathrm{~m}\) away? \((b) 3.0 \mathrm{~m}\) away? \((c) 1.0 \mathrm{~m}\) away? (d) What is the closest object this lens could photograph sharply?

Step-by-Step Solution

Answer

(a) Lens should be 105.47 mm in front. (b) Lens should be 110.16 mm in front. (c) Lens should be 118.33 mm in front. Closest object: 678.46 mm away.

Step 1: Understand the Lens Formula

The lens formula is given by \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\), where \(f\) is the focal length of the lens, \(d_o\) is the object distance, and \(d_i\) is the image distance. In this problem, \(f = 105 \text{ mm}\). We need to solve for \(d_i\) when \(d_o\) is given.

Step 2: Convert Object Distances

Since the object distances are given in meters, for consistency with the focal length in millimeters, convert these distances to millimeters. For Part (a), \(d_o = 10.0 \text{ m} = 10,000 \text{ mm}\). For Part (b), \(d_o = 3.0 \text{ m} = 3,000 \text{ mm}\). For Part (c), \(d_o = 1.0 \text{ m} = 1,000 \text{ mm}\).

Step 3: Solve for Image Distance in Part (a)

Substitute \(f = 105 \text{ mm}\) and \(d_o = 10,000 \text{ mm}\) into the lens formula: \(\frac{1}{105} = \frac{1}{10,000} + \frac{1}{d_i}\). Solving for \(\frac{1}{d_i}\) gives:\[\frac{1}{d_i} = \frac{1}{105} - \frac{1}{10,000} \]Calculate \(d_i\) after finding the value of \(\frac{1}{d_i}\).

Step 4: Solve for Image Distance in Parts (b) and (c)

Repeat Step 3 for the other object distances. For Part (b), substitute \(d_o = 3,000 \text{ mm}\). Solve \(\frac{1}{d_i} = \frac{1}{105} - \frac{1}{3,000}\). For Part (c), substitute \(d_o = 1,000 \text{ mm}\). Solve \(\frac{1}{d_i} = \frac{1}{105} - \frac{1}{1,000}\).

Step 5: Determine Closest Object for Focusing

The maximum distance allowed between the lens and the sensor is \(132 \text{ mm}\). Thus, \(d_i = 132 \text{ mm}\). Solve the lens equation for \(d_o\):\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]Substitute \(d_i = 132\) and \(f = 105\), and solve for \(d_o\).

Key Concepts

Focal Length

Focal length is a crucial concept when it comes to using lenses in optics. It is represented by the symbol \( f \) in equations. The focal length of a lens is the distance from the center of the lens to the point where it focuses parallel rays of light, known as the focal point. In the context of lenses, the focal length is typically measured in millimeters.

When dealing with camera lenses, the focal length determines the field of view:

• A shorter focal length lens captures a wider view, suitable for landscape photography.
• A longer focal length lens offers a narrow field of view, ideal for capturing distant subjects.

For the lens in the exercise, the given focal length is 105 mm. This value plays a critical role when applying the lens formula to find image or object distances in photography or other optical applications.

Image Distance

The image distance, denoted by \( d_i \), is the distance between the lens and the image formed on a particular plane, such as a camera sensor. This concept is central to determining how to properly focus a lens on a specific object.

In our exercise, the image distance is found using the lens equation:

• The lens equation relates the focal length \( f \), the object distance \( d_o \), and the image distance \( d_i \).
• It's solved as \( \frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} \).

Once you have the values of \( f \) and \( d_o \), you can calculate \( d_i \) to determine how far the lens should be from the camera's sensor to capture a sharp image. In our problem, various \( d_o \) values are provided, resulting in different required image distances. The maximum allowable \( d_i \) is given as 132 mm, which defines the closest possible focus point for the camera.

Object Distance

The object distance, symbolized as \( d_o \), refers to the distance from the object being photographed or focused on to the lens. This distance plays a significant role in the lens formula and helps us calculate how the lens should be adjusted to accurately showcase an image on the sensor.

In the problem, you're given multiple object distances: 10.0 m, 3.0 m, and 1.0 m. To align with the unit of focal length (in millimeters), these distances are converted to millimeters:

• 10.0 m becomes 10,000 mm
• 3.0 m converts to 3,000 mm
• 1.0 m changes to 1,000 mm

With these values, we can apply the lens equation to find out how the lens should be positioned for different photography scenarios. Accurately determining the object distance allows photographers to appropriately set their camera to capture clear images, depending on the distance of the subject.

Lens Equation

The lens equation is the mathematical formula that links the focal length, object distance, and image distance in lens systems. It is expressed as:\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]The equation essentially balances these three critical factors to ensure that light passing through the lens forms a focused image on the photographic sensor or another designated plane.

In practical applications:

• The formula allows you to solve for one unknown if the other two parameters are known.
• This makes it an indispensable tool in designing optical systems and in everyday photography.

In the given exercise, this equation is instrumental in calculating the correct image distance for varying object distances, thus allowing photographers to adjust the lens for optimal focus. For example, knowing the maximum \( d_i \) helps in determining the nearest distance at which clear photos can still be taken.