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

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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 . A_o 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/(A_oa²) and x=−a/T, the equation I = A_o T² e^-W/kT simplifies:

0 = e^x − 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).