I'm starting to work with simple circuits (learning how to measure electromagnetic variables). I have been looking up on the internet for a formula or deduction of a formula that relates the current $I$ and the voltage $V$ in a filament lamp, but haven't succeeded.
I did the experiment of measuring these variables and got something similar to a natural logarithm, but of course there should be some other constants in the equation $I(V)$. Do you know where can I find such an equation?
Best Answer
Resistive Filament - Voltage vs. Current
If you referring to measuring the $V(I)$ dependence of the heated filament I would derive an approximate equation as follows.
$T: \textrm{Temperature of the filament}$
$P_{dis} \equiv V \cdot I: \textrm{Power dissipated}$
$k: \textrm{Coefficient of temperature vs power}$
$R \equiv V/I : \textrm{Ohmic Resistance}$
$\alpha: \textrm{temperature coefficient of resistance}$
with
$$T = k P_{dis} = k I^2R$$
Now using a linear approximation for the resistance over temperature we have:
$$R = R_o[1+\alpha(T - T_o)] = R_o[1+\alpha k I^2 R - \alpha T_o]$$
Rearranging terms we have:
$$R[1 - R_o \alpha k I^2] = R_o[1 -\alpha T_o]$$
By definition we can substitute $R=V/I$ to get
$$V(I) = I \cdot R_o \cdot \left( \frac{ 1-\alpha T_o}{1-R_o \alpha k I^2} \right)$$
I have not tested this equation though. Note that you have one parameter that you can measure directly ($R_o$), one parameter you can look up ($\alpha$) and one that is unknown ($k$) because it depends on many factors including the shape of the wire. And I doubt that the equation will be accurate over a wide temperature range.
Thermionic Emmission of Electrons
If you are looking for the emission of electrons from a heated filament then I think you need the Richardson or Sommerfeld equations (http://en.wikipedia.org/wiki/Thermionic_emission) which are discussed more in depth here (http://www.tubebooks.org/Books/chaffee.pdf) on page 59. I don't think this is what you mean though.