The focal length of a convex mirror whose radius of curvature is 32 cm is 16 cm.
Understanding Focal Length and Radius of Curvature
For any spherical mirror, whether convex or concave, there's a direct and simple relationship between its focal length and its radius of curvature. The focal length ($f$) is precisely half of the radius of curvature ($R$). This relationship is fundamental in optics and is expressed by the formula:
$f = R / 2$
Applying the Principle to Convex Mirrors
A convex mirror is a spherical mirror that curves outwards, causing incident light rays to diverge. Despite this divergence, the same relationship between focal length and radius of curvature holds true. The focal point of a convex mirror is a virtual point located behind the mirror, where the diverging reflected rays appear to originate from. By convention in many optical calculations, the focal length of a convex mirror is considered positive, while its radius of curvature is also taken as positive when measured from the pole to the center of curvature.
Let's calculate the focal length for the given convex mirror:
- Radius of Curvature (R): 32 cm
Using the formula $f = R / 2$:
- $f = 32 \text{ cm} / 2$
- $f = 16 \text{ cm}$
Summary of Values
To summarize the calculation:
| Property | Value | Unit |
|---|---|---|
| Radius of Curvature | 32 | cm |
| Focal Length | 16 | cm |
Practical Applications of Convex Mirrors
Convex mirrors are widely used due to their ability to provide a wider field of view compared to plane mirrors or concave mirrors. This characteristic makes objects appear smaller but covers a larger area, which is beneficial in various applications:
- Vehicle Rearview Mirrors: Many passenger-side rearview mirrors are convex, allowing drivers to see a broader area behind their vehicle, though objects appear farther away than they actually are.
- Security Mirrors: Used in stores, parking garages, and offices, these mirrors help security personnel or shoppers monitor blind spots and large areas.
- Road Intersections: Placed at sharp bends or intersections with limited visibility, convex mirrors help drivers see oncoming traffic or pedestrians.
- ATMs: Small convex mirrors are sometimes integrated into ATM machines, allowing users to see if anyone is standing behind them, enhancing personal security.
Understanding the focal length is crucial for designing and utilizing optical instruments and systems effectively. For more details on spherical mirrors and their properties, you can refer to resources like Physics LibreTexts on Spherical Mirrors.
