A camera uses a lens with aperture \(2.0 \mathrm{cm} .\) What is the angular resolution of a photograph taken at \(700 \mathrm{nm}\) wavelength? Can it resolve the millimeter markings of a ruler placed 35 m away?

The Rayleigh criterion is the threshold at which we can distinguish two light sources as separate entities. It states that two sources are resolvable when the central maximum of the diffraction pattern from one source coincides with the first minimum of the diffraction pattern from the other source. In simpler terms, it helps us understand the limit at which optical instruments like microscopes, telescopes, or cameras can distinguish fine details.

When applied to our exercise, the Rayleigh criterion is used to calculate the angular resolution of a camera lens. It relies on the wavelength of light that the camera is sensitive to, and the diameter of the camera's lens to determine how closely two points can be to each other while still being perceived as distinct.

The wavelength of light is the distance between consecutive peaks or troughs in a light wave. It is usually measured in meters (m) and is denoted by the Greek letter \(\text{λ}\). Light with different wavelengths appears to us as different colors. In the context of angular resolution, the wavelength is crucial since it affects how much an optical system diffuses the light.

In the calculation provided, the wavelength of light used by the camera lens is \(700 \text{nm}\), which falls into the red portion of the visible spectrum. The shorter the wavelength, the better the angular resolution, meaning the instrument can distinguish finer details. Conversely, longer wavelengths will result in a lesser ability to resolve small features.

Lens diameter, denoted by \(D\), is a key factor in determining the angular resolution of optical instruments. It refers to the size of the lens's aperture, through which light enters. A larger lens diameter allows more light to be collected, which generally improves the resolution and detail that can be discerned in the image.

The diameter of the lens in our exercise is given as \(2.0 \text{cm}\) or \(0.02 \text{m}\). With a greater diameter, the denominator in the Rayleigh criterion formula increases, resulting in a smaller value for angular resolution \(\delta\). This indicates a higher resolution, allowing the instrument to resolve smaller or closer points. Here, the crucial relationship between lens diameter and resolution can be clearly seen: larger lenses are capable of resolving finer details compared to smaller lenses.