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The constant A in the Richardson-Dushman equation for tungsten is 60 × 10^4 A/m^2-K. The work function of tungsten is 4.5 eV. A tungsten cathode having a surface area 2.0 × 10^-5 m^2 is heated by a 24 W electric heater. In steady state, the heat radiated by the cathode equals the energy input by the heater and the temperature becomes constant. Assuming that the cathode radiates like a blackbody, calculate the saturation current due to thermions. Take Stefan constant = 6 × 10^-8 W/m^2-K4. Assume that the thermions take only a small fraction of the heat supplied.
Sol. A = 60 * 10^4 A/m^2 – k^2 ∅ = 4.5 eV σ = 6 * 10^–8 ω/m^2 – k^4 S = 2 * 10^–5 m^2 K = 1.38 * 10^–23 J/K H = 24 ω’ The Cathode acts as a black body, i.e. emissivity = 1 ∴ E = σ A T^4 (A is area) ⇒ T^4 = E/σA = 24/6 * 10^-8 *2 * 10^-5 = 2 * 10^13K = 20 *10^12K ⇒ T = 2.1147 * 10^3 = 2114.7 K Now, i = AST^2 e^-∅/KT = 6 * 10^5 * 2 * 10^-5 * (2114.7)^2 * e^-4.5*1.6*10^-19/1.38*T*10^23 = 1.03456 * 10^-3 A = 1 mA
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