Question

approximately what is the smallest detail observable with a miscrope that uses red light of freuqency 4.32 x 10¹⁴ hz?

Answer 1

The smallest detail observable with a microscope using red light of frequency
`\(4.32 * 10^(14)\)` Hz is approximately 389 nanometers, determined by diffraction limits.

The resolution of an optical microscope is limited by the diffraction of light, described by the Rayleigh criterion. The minimum resolvable detail (smallest resolvable feature) can be estimated using the formula

`\( \text{Resolution} \approx \frac{\lambda}{2\text{NA}} \)`, where
`\( \lambda \)` is the wavelength of light and NA
(numerical aperture) characterizes the lens.

For red light with a frequency of
`\(4.32 * 10^(14)\)` Hz, we can use the speed of light equation
`(\(c = f \lambda\))` to find the wavelength

`(\( \lambda = (c)/(f) \))`. Assuming red light has a wavelength around 700 nm, or
`\(7 * 10^(-7)\)` meters, and an NA of around 0.9 for a good microscope lens, we get:

`\[ \text{Resolution} \approx (7 * 10^(-7))/(2 * 0.9) \]`

Calculating this gives an approximate resolution of
`\(3.89 * 10^(-7)\)` meters or 389 nm. Therefore, the smallest detail observable is around 389 nanometers.