Question

Focal Length and Power (for Multiple Lenses in Contact)

Answer 1

The focal length (f) of lenses in contact can be determined using the formula for combining lenses in contact:
`\( (1)/(f) = (1)/(f_1) + (1)/(f_2) \), where \( f_1 \) and \( f_2 \)` are the focal lengths of the individual lenses.

Step-by-step explanation:

When two lenses are in contact, their combined focal length is calculated using the lensmaker's equation:
`\( (1)/(f) = (n - 1) \left( (1)/(R_1) - (1)/(R_2) \right) \),` where \( f \) is the focal length and
`\( R_1 \) and \( R_2 \)` are the radii of curvature of the lens surfaces. This equation is used for each lens, resulting in two focal lengths
`(\( f_1 \)` and
`\( f_2 \)).` Substituting these values into the formula
`\( (1)/(f) = (1)/(f_1) + (1)/(f_2) \)` provides the combined focal length of the lenses in contact.

The formula demonstrates the reciprocal relationship between focal length and lens power. As the focal length decreases, the lens power increases, and vice versa. This relationship is critical in optics, as it allows for the manipulation of light to achieve desired outcomes in various optical systems. Additionally, understanding the combined focal length of lenses in contact is fundamental for designing optical devices such as eyeglasses, microscopes, and telescopes, where multiple lenses work together to form the desired image.

In conclusion, the relationship between focal length and lens power is encapsulated in the formula
`\( (1)/(f) = (1)/(f_1) + (1)/(f_2) \)` when dealing with lenses in contact. This formula enables optical engineers and designers to predict and manipulate the combined focal length of multiple lenses, shaping the behavior of light in optical systems.