Question

{"given":"Work function $\\phi = 4.0$ eV. The longest (threshold) wavelength $\\lambda_0$ satisfies: $$\\phi = \\frac{hc}{\\lambda_0}$$","key_observation":"At the threshold wavelength, the photon energy exactly equals the work function. Using the standard relation $hc = 1240$ eV·nm, the threshold wavelength is $\\lambda_0 = \\dfrac{1240}{\\phi}$ nm.","option_analysis":[{"label":"(A)","text":"540 nm","verdict":"incorrect","explanation":"Photon energy $= 1240/540 \\approx 2.3$ eV, which is less than $\\phi = 4.0$ eV, so this wavelength cannot cause photoemission."},{"label":"(B)","text":"400 nm","verdict":"incorrect","explanation":"Photon energy $= 1240/400 = 3.1$ eV, which is still less than $\\phi = 4.0$ eV, so photoemission cannot occur."},{"label":"(C)","text":"310 nm","verdict":"correct","explanation":"$\\lambda_0 = \\dfrac{1240}{4.0} = 310$ nm. This photon energy exactly equals the work function, making it the threshold (longest) wavelength for photoemission."},{"label":"(D)","text":"220 nm","verdict":"incorrect","explanation":"Photon energy $= 1240/220 \\approx 5.6$ eV $> \\phi$, so photoemission occurs but this is not the longest (threshold) wavelength."}],"answer":"(C)","formula_steps":[]}