91Ó°ÊÓ

In optics, the lens formula is a crucial equation that helps us relate key parameters of a lens-based system. These parameters include the focal length (\( f \)), object distance (\( d_o \)), and image distance (\( d_i \)). The formula is given by:

• \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \)

This equation is derived from the geometry of lenses and the way they bend light. It essentially reflects how the curvature and position of an object impact where its image will form relative to the lens.
By rearranging this formula, you can solve for any one of these three variables if the other two are known. This makes it a fundamental tool in solving various optics problems, such as determining how much you need to move a camera lens when focusing on objects at different distances.

The focal length of a lens is a measure of how strongly the lens converges or diverges light. It is the distance from the lens to the point where parallel rays of light converge to a single point (focus). In our exercise with the camera, the focal length given is 90 mm.
Shorter focal lengths mean the lens has a stronger ability to bend light, which is useful for creating wider fields of view.

• Positive focal length indicates a converging lens, which brings light to a point.
• Negative focal length is typical for diverging lenses, spreading light apart.

Focal length is key in determining how far away you must be from a subject to bring it into clear focus. It also influences the size of the image formed on the film or sensor and is crucial for calculating both image and object distances using the lens formula.

Image distance, denoted as \( d_i \), is the distance from the lens to the point where the image is formed. This is a critical factor in determining the clarity and scale of the image captured by a camera.
Using the lens formula, we calculate image distances for both the initial and new focus positions in our problem. Initially with an object distance of 1.30 m, the image distance calculated was approximately 97.03 mm. With a change in the object distance to 6.50 m, the image distance shifted to 90.73 mm.
The change in image distance (\( \Delta d_i \)) can quantitatively show how the lens should be relocated within a camera apparatus. In this case, we determined the lens would need to move 6.30 mm closer to maintain focus on the object.

Object distance, usually symbolized as \( d_o \), is the essential distance between an object and a lens. This distance impacts where the image will form relative to the lens and thus is central to adjusting the focus.

• In the initial scenario, our object was at 1.30 m (or 1300 mm) from the lens.
• Refocusing was required for a new object distance of 6.50 m (or 6500 mm).

Changes in object distance directly affect the image distance, as evident from the change in focus involving moving the lens. Grasping this relationship allows photographers and optical engineers to manipulate lenses suitably for sharp imaging, efficiently using the lens formula to adjust focus settings. Understanding this interplay is central to mastering optics and ensuring clear, focused images.