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The resistivity of Tungsten is given by \$\rho(T) \propto T^{1.209}\$ (from Paul Gluck's Physics Project Lab).

Let's assume that we can ignore the changes in the geometry of the wire due to temperature change.

Does that mean that for a Tungsten filament with applied voltage \$V\$ and current \$I\$, we have \$V/I\propto T^{1.209}\$ or \$dV/dI\propto T^{1.209}\$?

Note that \$V(I)\$ is not linear, so the distinction is relevant.

As I understand it, the former is resistance, and the latter is differential-resistance. Which one of them is related to temperature as given above?

Rd Basha
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2 Answers2

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The resistivity, \$\rho(T)\$, is temperature-dependent. \$T\$ is the material temperature here.

Which one of them is related to temperature as given above?

Neither.

Resistance is

$$ R=\rho \ \frac{l}{A} $$

where \$l\$ and \$A\$ are length and cross-sectional area, respectively.

The only clear thing is \$\rho(T)\propto T^{1.209}\$. But it's unclear if the physical dimensions are temperature-dependent as well. So we may not say \$R(T)\propto T^{1.209}\$ instantly.

If you apply a voltage of \$V\$ across a temperature-dependent resistance of \$R(T)\$, a current of \$I\$ flows.

$$ V/I=R(T)=\rho(T) \ \frac{l}{A} $$

If we assume the physical dimensions remains unchanged with temperature then we can write \$R(T)\propto T^{1.209}\$ so \$V/I\propto T^{1.209}\$. But this is a result of some assumptions.

The relation between the dissipation of \$P=V \ I\$ and the final material temperature is not given, or not clear, or not included. So if you increase the applied voltage by \$dV\$ the resistance change due to the dissipation increase may or may not be proportional to \$T^{1.209}\$.

Rohat Kılıç
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\$\rho\$ is the resistivity, resistance when given a geometry, that relates the V and I that you measure at the terminals of the tungsten wire.

If you want the dV/dI to get differential resistance, then you need the slope of the VI curve.

You can estimate a VI curve given \$\rho(T)\$, making some other assumptions which have varying degrees of validity at various temperatures, like radiation dominates so heat lost to connections and convection is negligible, emmissivity is constant and Stefan's Law holds. In general though it's best to measure the VI curve for any particular lamp construction.

Whether the VI curve is linear or not depends on the frequency of the V. If the frequency is high, then there's no time for the temperature and hence resistance to change, and so the lamp behaves as a linear resistor. This effect is put to good use in lamp-stabilised Wein Bridge oscillators, and substitution RF power-meters.

Neil_UK
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