Question

A physics student is eying a steel drum, the top part of which has the approximate shape of a concave spherical surface. The surface is sufficiently polished that she can just barely make out the reflection of her finger when she places it above the drum. As she slowly moves her finger toward the surface and then away from it, you ask her what she is doing. She replies that she is estimating the radius of curvature of the drum. How can she do that?

Short answer

Answer: The student can estimate the radius of curvature by finding the position where her finger and its reflected image appear to overlap. At this point, the object distance (her finger) and image distance are equal. Using the mirror formula and the relationship between the radius of curvature and focal length, the radius of curvature is equal to the object or image distance. The student can measure this distance to estimate the radius of curvature.

Step-by-step solution

Find the object and image distances

When the student moves her finger toward the surface and the reflected image appears to overlap her fingertip, the object distance (distance from the surface to the finger) and image distance (distance from the surface to the image) are equal.

Use the mirror formula

The mirror formula is given by: \[\frac{1}{f} = \frac{1}{u} + \frac{1}{v}\] Where f is the focal length, u is the object distance, and v is the image distance. Since u = v, the formula simplifies to: \[\frac{1}{f} = \frac{2}{u}\]

Relate the focal length and radius of curvature

In a spherical mirror, the relationship between the radius of curvature and the focal length is: \[R = 2f\] Where R is the radius of curvature.

Calculate the radius of curvature

Now we can use the derived equation from step 2, for the focal length as a function of the object distance: \[f = \frac{u}{2}\] And substitute this relationship into the equation to find the radius of curvature: \[R = 2 \left(\frac{u}{2}\right) = u\] So when the object distance is equal to the image distance, the radius of curvature is equal to the object(distance between her finger and the surface) or image distance. The student can measure this distance to estimate the radius of curvature of the drum's concave surface.

Key concepts

Radius of Curvature

The radius of curvature (R) is a fundamental concept in the study of spherical mirrors. It is defined as the radius of the sphere from which the mirror segment was cut.
Imagine a large shiny ball cut into pieces; the arc of each piece, representative of a spherical surface, has a curvature with a radius identical to that of the original ball.
Specifically for a concave mirror, as seen in the physics student's experiment, radius of curvature is a measure from the mirror's surface to the center of that imaginary sphere. When dealing with simple geometrical optics, this measurement is crucial because it helps to determine the behavior of light interacting with the mirror.

Focal Length

Focal length (f) refers to the distance between the mirror surface and the focal point, where light rays either converge or appear to diverge. For a concave mirror, the focal point is real and situated in front of the mirror, where light rays converge. Conversely, for a convex mirror, the focal point is virtual and located behind the mirror, as reflected rays diverge.
Understanding focal length is vital as it determines the mirror's ability to magnify objects or produce an image. In the scenario of the steel drum, the focal length is directly related to the radius of curvature by the equation:
\[ R = 2f \],
and gauging one allows the student to deduce the other.

Spherical Mirrors

Spherical mirrors are mirrors that have a curved surface like a portion of a sphere. They fall into two categories: concave mirrors and convex mirrors.
Concave mirrors, like the one exemplified by the steel drum's surface, curve inward and can form real images when the object is placed at a distance greater than the focal length of the mirror. These mirrors are often used in devices where image magnification or concentration of light is required.
Convex mirrors, on the other hand, curve outward and only produce virtual images as they diverge light rays. Convex mirrors are typically used for wider field of vision applications, such as rearview mirrors in vehicles.

Object Distance

Object distance (u) in the context of mirror optics, denotes the distance between the mirror itself and the object being reflected. This variable is a key element in constructing ray diagrams and for calculating the properties of the image produced.
In the exercise, the student finds the point where her finger's reflection aligns perfectly with the finger itself—a critical observation. At this juncture, the object distance becomes equal to the image distance, revealing that the finger is situated at the center of curvature of the mirror, a span equal to twice the focal length from the vertex of the mirror.

Image Distance

Image distance (v) is the counterpart to object distance. It indicates how far the image is from the mirror's surface. If the image formed is real, as it is in the case of a concave mirror when the object is beyond the focal point, this distance is taken to be positive. In other situations, such as with virtual images formed by convex mirrors, the image distance is negative, indicating the image is behind the mirror.
In the case presented in the textbook exercise, when the image and object distances are equal, and knowing the relationship between the radius of curvature and focal length, the student cleverly deduces the radius of curvature by measuring this distance. This insightful trick allows for an estimation of the curvature of the drum without requiring sophisticated tools.