You wish to construct a pinhole camera (no lens) with \(10 \mathrm{~cm}\) separation between pinhole and film plane. The film is sensitive to visible light \((\lambda \sim 500 \mathrm{~nm}) .\) What pinhole diameter would you choose for optimum resolution, i.e., smallest photographic image of a point object, and what order of magnitude of angular resolution would be obtained?

The Rayleigh criterion states that when the central maximum of the diffraction pattern of one light source overlaps with the first minimum of the diffraction pattern of another light source, the two sources are barely resolved. The Rayleigh criterion formula for a pinhole camera is given by: \(θ_{\text{min}} = 1.22 * \frac{\lambda}{D}\), where \(θ_{\text{min}}\) is the minimum angular separation between two light sources, \(\lambda\) is the wavelength of light (here, visible light is around 500 nm), and \(D\) is the pinhole diameter. In this case, we'll need to rearrange this equation to solve for \(D\).

Rearranging the Rayleigh criterion formula to solve for the pinhole diameter \(D\): \(D = \frac{1.22 * \lambda}{θ_{\text{min}}}\) Since we want the smallest possible image of a point object, we assume that the minimum angular separation is equal to the angular resolution. In other words, \(θ_{\text{min}} = θ_{\text{res}}\).

We now have the formula to calculate the optimal pinhole diameter for the given camera configuration. Given a 10 cm distance between the pinhole and the film plane, the optimal diameter is given by: \(D = \frac{1.22 * \lambda}{θ_{\text{res}}}\) Plug in the given value of \(\lambda = 500 \mathrm{~nm}\): \(D = \frac{1.22 * (500 \times 10^{-9} \mathrm{~m})}{θ_{\text{res}}}\) To find the optimal pinhole diameter, we'll need to minimize the value of \(θ_{\text{res}}\). Since \(\frac{d}{\mathrm{tan}(θ)}\) represents the linear size of an image (where d is the distance between the pinhole and the film plane), the best resolution occurs when the linear size of the image has a minimum value, i.e., \(Δx = θ_{\text{res}} \times d\) is minimum. Taking the derivative of \(Δx\) with respect to \(θ_{\text{res}}\) and setting it to zero, we get: \(\frac{d[θ_{\text{res}} \times d]}{dθ_{\text{res}}} = d = 0\) This implies that \(θ_{\text{res}} = \frac{1.22 * \lambda}{D} \approx \frac{\lambda}{D}\) is minimum in our situation. Therefore, we get: \(D = 1.22 * \lambda = 1.22 * (500 \times 10^{-9} \mathrm{~m})\) Now, calculate the value of \(D\): \(D \approx 6.1 \times 10^{-7} \mathrm{~m}\) So the optimal pinhole diameter is approximately \(6.1 \times 10^{-7} \mathrm{~m}\).

Now that we know the optimal pinhole diameter, we can calculate the order of magnitude of angular resolution. From the Rayleigh criterion formula, we have: \(θ_{\text{res}} = \frac{1.22 * \lambda}{D}\) Plug in the known values of \(\lambda = 500 \mathrm{~nm}\) and \(D = 6.1 \times 10^{-7} \mathrm{~m}\): \(θ_{\text{res}} = \frac{1.22 * (500 \times 10^{-9} \mathrm{~m})}{6.1 \times 10^{-7} \mathrm{~m}}\) Calculate the value of \(θ_{\text{res}}\): \(θ_{\text{res}} \approx 1 \times 10^{-3} \,\mathrm{radians}\) Thus, the order of magnitude of angular resolution is around \(1 \times 10^{-3}\) radians.