Final answer:
To calculate the focal length of a lens needed to focus a helium-neon laser to a specific spot size, the diffraction limit formula must be used. Additional information such as the diameter of the lens aperture is necessary to solve for the focal length. Without it, the problem cannot be accurately resolved.
Step-by-step explanation:
A scientist needs to focus a helium-neon laser beam (λ=633nm) to a 10-μm-diameter spot 8.5 cm behind the lens. To find the appropriate focal length of the lens, one can use the formula for the diffraction limit of a circular aperture, which is given as: θ = 1.22 × (λ / D) Where θ is the angular spread of the light, λ is the wavelength of the laser light, and D is the diameter of the lens aperture. However, since the question does not provide the diameter of the lens, we instead relate the size of the spot directly to the focal length using the following equation which arises from the diffraction limit for a lens: s = f × θ If we take the spot size 's' to be equivalent to the diameter of the focused spot which is 10 μm, set the focal length 'f' as what we're solving for, and include the known value 's' and calculated θ from the diffraction limit equation, we can solve for 'f'. But to find θ, we need to consider that it should be small enough such that when multiplied by the focal length 'f', it yields a spot of 10 μm diameter θ = 1.22 × (λ / D) = 1.22 × (633e-9 m) Without the lens diameter, we cannot proceed further. Typically in problems like this, we would use the size of the aperture (if known) to derive the minimum angular spread, and from there find the focal length required for the specific spot size. Thus, additional information, namely the aperture size of the lens, is required to accurately determine the focal-length lens needed to create a 10-μm-diameter spot with a helium-neon laser.