Temperature distribution in an electrically heated filament

Abstract

The distribution of temperature in a filament electrically heated in vacuo has been studied by several previous authors. The differential equation defining the steady state is: d²T/dx² + 5/2a (Tm⁴-T⁴) = 0, in which T is the temperature at a distance x from one of the ends, T_m is the value to which the temperature T_L at the centre tends as the length 2l of the filament is increased indefinitely, keeping the heating current the same, and a is a constant determined by the cross-section of the filament, its thermal conductivity and the emissivity of its surface. Using the boundary conditions T = θ when x = 0, and dT/dx = 0 when x = l, one obtains x = ∫^T_θ [5aTm⁴(T₁-T) -a(T₁⁵- T⁵]- ⁱ dT, (2) the value of T_l occurring in (1) being determined by the condition that, when the upper limit of the integral is made equal to T_l, x should become l.