Wikipedia:Reference desk/Archives/Mathematics/2014 April 21

= April 21 =

Inverse Richardson-Dushman equation
Textbooks on thermionic emission give an equation for finding the saturation current (I) for a given temperature (T) :-
 * I = A . Ao T2 e-W/kT

How do I invert this to find T when I is known? 121.221.225.6 (talk) 13:28, 21 April 2014 (UTC)
 * The Lambert W function can be used to give a closed form solution, but really your best bet is to plot the function and get the inverse from the plot. Or you could write a program that does a binary chop on an interval and looks at the values till you get as close as you like. Dmcq (talk) 17:27, 21 April 2014 (UTC)

Letting a=W/k and b=I/(Aoa2) and x=−a/T, the equation I = Ao T2 e-W/kT simplifies:
 * 0 = ex − bx2

Expand the exponential:
 * 0 = 1 + x + x2(1/2−b) + x3/6 + x4/24 + ⋅ ⋅ ⋅

When the series is truncated to a polynomial the resulting algebraic equation is solved by some numerical root-finding algorithm. Bo Jacoby (talk) 18:28, 21 April 2014 (UTC).