For tungsten, the constant A in the Richardson–Dushman equation is 60 imes 10^4 A m ⁻² K ⁻² , and its work function is 4.5 eV . An electric heater of 24 W is used to heat a tungsten cathode with a surface area of 2.0 imes 10⁻⁵ m ^2 . In the steady state, the temperature remains constant as the heat radiated by the cathode matches the energy input from the heater. Assuming the cathode radiates as a perfect blackbody,

Step-by-step solution

In the steady state, the power supplied by the heater is equal to the power radiated by the cathode as a blackbody. P = S T^4 Substituting the given values: 24 = (6 10⁻⁸) (2.0 10⁻⁵) T^4 24 = 12 10⁻¹³ T^4 T^4 = 24 12 10⁻¹³ = 2 10¹³ = 20 10¹² K ^4 T^2 = 20 10^6 4.472 10^6 K ^2 T = 4.472 10^3 2115 K Now, we calculate the exponent kT for the Richardson-Dushman equation: kT = 4.5 1.6 10⁻¹⁹ 1.38 10⁻²³ 2115 7.2 10⁻¹⁹ 2.918 10⁻²⁰ 24.67 The exponential factor is: e^ - / kT = e^ -24.67 1.93 10⁻¹¹ The saturation current I is