Focal Length and Radius of Curvature

If we look at a spherical mirror, the relationship between its focal length (f) and radius of curvature (R) can be expressed as f=R2, where f is half the radius of curvature and R is the focal length.

Focal Length

The focal length of an optical system is a measure of how strongly the system converges or diverges light; it is the inverse of the system’s optical power. The focal length of an optical system is the distance between two points in space. An increase in focal length indicates that a system is bringing light closer together, whereas a decrease in focal length indicates that the system is diverging light. Because of the shorter focal length, the rays are bent sharper, bringing them closer together or diverging more quickly as they pass through the system. An example of a positive focal length is the distance over which initially collimated (parallel) rays are brought to a focus; an example of a negative focal length is the distance in front of the lens over which a point source must be located in order to produce a collimated beam. If we consider more general optical systems, the focal length has no intuitive significance; it is simply the inverse of the optical power of the system.

Radius of Curvature

The radius of curvature, denoted by the letter R. It is equal to the radius of the circular arc that most closely approximates the curve at that point in time for a curve. The radius of curvature of a surface is defined as the radius of a circle that best fits a normal section or combinations of normal sections.

In order to denote the optical radius of curvature, the following sign convention is used:

It is possible for the radius of curvature to be positive if the vertex is located to the left of the centre of curvature.

It is possible for the radius of curvature to be negative if the vertex is located to the right of the centre of curvature.

As a result, when looking at a biconvex lens from the side, the radius of curvature of the left surface is positive, while the radius of curvature of the right surface is negative.

It should be noted, however, that other sign conventions are occasionally employed in areas of optics other than design. A common example is the Gaussian sign convention, which states that convex surfaces of lenses are always positive. Many undergraduate physics textbooks follow this convention. When combining formulas from various sources, caution should be exercised.

The Relationship Between the Focal Length and the Radius of Curvature

An inward or outward curve can be used to shape the reflecting surface of a sphere mirror, depending on the application. In optics, the focal length of a lens is denoted by the letter f. Curvature is represented by the letter r, and it is defined as the radius of a lens that can be used to form an entirely round object.

In this case, the incident light, which is parallel to the principal axis and incident on a spherical mirror at point D, is considered. The radius of curvature is represented by CP and CP = R. As a result of reflection , the light will pass through the focus of concave mirror F or will appear to diverge from the convex mirror, and will obey the law of reflection, I = r.

From the geometry of the figure,

We can quickly determine the relationship.

 f = R/2

This type of relationship can be used to describe convex mirrors as well. In this relationship, it is assumed that the mirror’s aperture is relatively small.

When using spherical mirrors with small apertures, it has been observed that the radius of curvature is equal to twice the focal length. As a result, R = 2f.

We can state unequivocally that the primary focus of a spherical mirror is located in the middle of the mirror, between the centre of curvature and the pole.

Conclusion

The focal length is represented by the letter f, and the radius of curvature is represented by the letter ‘R.’ The focal length of a mirror is defined as the distance between the focus (F) and the pole (P). The distance between the focus F and the centre of curvature (C) is referred to as the radius of curvature, where R denotes the radius of curvature and C denotes the centre of curvature.

If we look at a spherical mirror, the relationship between its focal length (f) and radius of curvature (R) can be expressed as f=R/2, where f is half the radius of curvature, and R is the radius of curvature.