The focal length of a mirror with a radius of curvature of 20 cm is 10 cm.
Understanding Focal Length and Radius of Curvature
To understand this calculation, it's essential to define two key properties of spherical mirrors:
- Focal Length (f): This is the distance from the mirror's pole (the center of its reflective surface) to its principal focus. The principal focus is the point where parallel rays of light converge after reflection from a concave mirror or appear to diverge from after reflection from a convex mirror.
- Radius of Curvature (R): This is the radius of the sphere of which the mirror is a part. It is the distance from the mirror's pole to its center of curvature (the center of that sphere).
The Fundamental Relationship: f = R/2
For spherical mirrors, there is a direct and fundamental relationship between the focal length and the radius of curvature. For both concave and convex mirrors, the focal length is exactly half of the radius of curvature:
$$ f = \frac{R}{2} $$
This relationship holds true for paraxial rays (rays close to and parallel to the principal axis of the mirror), which is a common approximation in introductory optics.
Calculation for the Given Mirror
Given a mirror with a radius of curvature ($R$) of 20 cm, we can calculate its focal length ($f$) using the formula:
- Given: Radius of Curvature ($R$) = 20 cm
- Formula: $f = R / 2$
- Calculation: $f = 20 \text{ cm} / 2 = 10 \text{ cm}$
Therefore, the focal length of the given mirror is 10 cm.
Properties of Spherical Mirrors
The sign convention for focal length and radius of curvature typically depends on the type of mirror:
| Feature | Concave Mirror | Convex Mirror |
|---|---|---|
| Shape | Reflective surface curves inward | Reflective surface curves outward |
| Focal Length ($f$) | Positive | Negative |
| Radius of Curvature ($R$) | Positive | Negative |
| Light Rays | Converge parallel rays | Diverge parallel rays (appear to originate from focus) |
| Applications | Shaving mirrors, reflecting telescopes, solar furnaces | Rearview mirrors in vehicles, security mirrors, street light reflectors |
Practical Insights
Understanding the focal length and radius of curvature is crucial for designing and using optical instruments.
- Magnification: The focal length determines how much a mirror will magnify or demagnify an object.
- Image Formation: The position and nature of the image formed by a mirror (real or virtual, inverted or upright, magnified or diminished) depend on the object's position relative to the mirror's focal point and center of curvature.
- Applications:
- Concave mirrors (positive focal length) are used in applications where light needs to be focused or collected, such as in satellite dishes, solar cookers, and dentist mirrors for magnification.
- Convex mirrors (negative focal length) are used when a wider field of view is required, like in vehicle side mirrors or store security mirrors, as they always produce virtual, upright, and diminished images.
For more detailed information on spherical mirrors and their properties, you can refer to reputable physics resources like HyperPhysics or educational textbooks on optics.
