# When drift velocity equals thermal velocity?

In some papers, I can see the drift velocity of electrons equaling thermal velocity. Can anyone tell me when both almost equal each other?

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The drift velocity is the net velocity of electrons in a certain direction under an applied field. The thermal velocity is has no net direction because it is randomly distributed and occurs in any metal at finite temperatures. Since the two velocities are different, it does not make any sense to say they are qualitatively equal, even though you may equate them quantitatively as in the previous answer.

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Thermal velocity is due to thermal agitation, and drift velocity is what you get if you superimpose this with the voltage driven motion. It is possible that the two velocities can be of the same value. The question is not asking about qualitative identity, is asking when they can be numerically equal. My simple analysis answers just that. I don't know the significance of this or any effects this might have on the conductivity of a metal wire, but it is certainly a good question. –  JKL Mar 19 '13 at 22:40
It provides no meaningful insight whatsoever! –  mcodesmart Mar 26 '13 at 22:42

The drift velocity of electrons in a metal is given by the equation

$I=enAv_D$

where $I$ is the electric current in the metal wire, $n$ is the number of electron density, $A$ is the cross sectional area of the metal wire and $v_D$ is the drift velocity.

From this we get

$v_D= \frac{I}{enA}$

The thermal velocity is given by

$\frac{1}{2}m_ev_T^2=\frac{3}{2}k_BT$

where $T$ is the temperature of the metal, $k_B$ is Boltzmann’s constant, $m_e$ is electron mass and $v_T$ is the thermal velocity. From the last equation we get

$v_T= \sqrt{\frac{3k_BT}{m_e}}$

Equating $v_D$ and $v_T$ gives the following

Condition for the two speeds to be equal:

$\frac{I}{enA }=\sqrt{\frac{3k_BT}{m_e}}$.

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