A pure tungsten cathode and a thoriated-tungsten cathode share the same geometrical dimensions and are operated at a common temperature. The current produced by the thoriated-tungsten cathode is 5000 times greater than that of the pure tungsten cathode. Determine the operating temperature. For pure tungsten, the constant A in the Richardson–Dushman equation is 60 10^4 A m ⁻² K ⁻² , while for thoriated tungsten it is

Step-by-step solution

Using the Richardson-Dushman equation, the emission current is given by: I = S A T^2 e^ - kT For pure tungsten: I₁ = S A₁ T^2 e^ - ₁ kT For thoriated tungsten: I₂ = S A₂ T^2 e^ - ₂ kT Given that I₂ = 5000 I₁ , we get: S A₂ T^2 e^ - ₂ kT = 5000 S A₁ T^2 e^ - ₁ kT A₂ A₁ e^ ₁ - ₂ kT = 5000 Substituting the given values A₁ = 60 10^4 A m ⁻² K ⁻² , A₂ = 3.0 10^4 A m ⁻² K ⁻² , ₁ = 4.5 eV , and ₂ = 2.6 eV : 3.0 10^4 60 10^4 e^ 4.5 - 2.6 kT = 5000 1 20 e^ 1.9 eV kT = 5000 e^ 1.9 eV kT = 100000 = 10^5 Taking the natural loga