Photocell With the Wrong Window: Red Laser and UV Line
A student is testing a photocell with a zinc cathode of work function 4.3 eV. The cell has an ordinary glass window. A red laser of wavelength 650 nm produces no photocurrent even when its intensity is increased, while a mercury lamp with a strong line at 254 nm produces a clear current. The student concludes that the red laser is defective. A second photocell with a cesium cathode of work function 2.1 eV is available; it has a quartz window. The student now wants to use the 254 nm line and several other lines to determine Planck's constant from stopping-potential measurements. The red laser's power is 1.0 mW, and its beam diameter is 2.0 mm. The glass window transmits above about 350 nm but absorbs strongly below that. The student asks why the red laser fails and whether the zinc cell can be used for the 254 nm measurements.
1. Calculate the photon energy of the 650 nm red laser. Explain why increasing its intensity does not produce photocurrent from zinc.
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Using E = hc/λ = 1240 eV·nm / λ,
E_red = 1240/650 ≈ 1.91 eV.
The zinc work function is 4.3 eV. A single red photon carries only 1.91 eV, less than the minimum energy needed to eject an electron. No photocurrent can flow no matter how many such photons arrive. Increasing the laser intensity increases the number of photons per second, but each photon still has only 1.91 eV. The laser is not defective; the photon energy is simply below the threshold. This is a frequency (or wavelength) threshold effect, not an intensity effect.
2. For the 254 nm mercury line incident on zinc, calculate the maximum kinetic energy of the emitted electrons and the stopping potential.
Read the worked answer
The photon energy is
E = 1240/254 ≈ 4.88 eV.
Einstein's photoelectric equation gives K_max = E − φ = 4.88 − 4.3 = 0.58 eV.
The stopping potential V_s is defined by eV_s = K_max, so V_s ≈ 0.58 V. The measurement must use the correct polarity: the collector is made negative relative to the cathode, and the current falls to zero at about 0.58 V. This value is independent of the light intensity, which is why stopping potential is useful for determining Planck's constant.
3. Should the student use the zinc cell with its glass window for the 254 nm measurements? Recommend the better choice and calculate the stopping potential expected with the cesium cell.
Read the worked answer
The zinc cell is a poor choice for 254 nm because ordinary glass absorbs strongly below about 350 nm. Much or all of the 254 nm light may be absorbed before reaching the zinc cathode, so the measured current and stopping potential would be unreliable or zero. A quartz window is needed for UV work.
The cesium cell with the quartz window is better. For 254 nm on cesium, K_max = 4.88 − 2.1 = 2.78 eV, so the expected stopping potential is about 2.78 V. The student should verify that the quartz window transmits 254 nm and that the collector polarity is correct. Using several known wavelengths, a plot of V_s against 1/λ has slope hc/e, from which h can be obtained.