# Is electron velocity at induction higher than in a wire?

When looking to the electrostatic induction on a microscopic level, do the electrons really move with high velocities or they move like when a current passes through the wire (slowly).

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Electrons move with a very high velocity even in an ordinary current. It's just that they go in all directions. The net drift velocity is slow. – Michael Brown Mar 30 '13 at 10:03
@Michael: Hi Michael. Sorry again. But. (like I've already told you) your comments cover some keywords of an answer, which sometimes may prevent others from posting it ;-) – Waffle's Crazy Peanut Mar 30 '13 at 10:36

## ELECTRON VELOCITY IN A WIRE AND IN AN INDUCTANCE

It is not clear whether the person asking the question means to find the drift velocity in the wire and inductance in the same circuit, or in two separate circuits and comparing them. In the latter case there is an interesting effect that inductance has on the speed (drift velocity) of the electrons, and I thought to share it with those who might be interested.

All answers, here and in several other places in this forum, make reference to the fundamental equation giving the drift velocity

$v_d=\frac{I}{enA}\tag1$

where

$I$ is the electric current in the circuit (in the wire)

$e=\mathrm{1.6\times 10^{-19}C}$

$n$ is the electron number density ($m^{-3}$)

$A$ is the cross-sectional area of the wire ($m^2$).

Let us consider two distinct circuits:

(i) An ideal battery with emf=E and a resistor with resistance R. In this case $I=E/R$ and equation (1) for the drift velocity becomes

$v_d=\frac{E}{enAR}\tag2$

We see that the drift velocity is fixed at all times, in the context of the model.

(ii) Now we consider an ideal battery with emf=E, and an inductance with self induction coefficient $L$ and resistance R. Upon switching the circuit on, the current in the circuit will follow the transient time solution (not difficult to show)

$I(t)=\frac{E}{R} (1-\exp[-(R/L)t])$.

So the drift velocity varies with time as

$v_d(t)=\frac{E}{enAR}(1-\exp[-(R/L)t])\tag3$

Equation (3) shows that at t=0 the drift velocity is actually zero, and it is only for very large times $t$ that it becomes equal to the $v_d$ of equation (2) where there is no inductance in the circuit. Therefore, on average, the electrons in case (i) are moving faster than in case (ii).

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No... The velocity of electron is just the same As Michael says, the electrons experience an average velocity called drift velocity (and, yes - only in presence of an Electric field), by random collisions with atoms in the conductor. Current is just an effect - observed in opposite direction to the motion of electrons.

If you take the time between two successive collisions, then drift velocity $v_d$ can be related to the electron's acceleration as $a=v_d/t$. As the motion can be explained via Newton's law, $m\times v_d/t$. As this motion is provided by the electric field $E=F/e$, the drift velocity can simply be $$v_d=\frac{eE}{m}t$$

It can be easily found by the relation of it with the current $I=nAev_d$ and it's around $0.1$ mm/s. Whenever the electron moves in a conductor, it experiences the some drift velocity...

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Nice answer ;-) – Michael Brown Mar 30 '13 at 11:12
@MichaelBrown: Hope I didn't plagiarize anything from your comment :P – Waffle's Crazy Peanut Mar 30 '13 at 11:55
I don't have a copyright on drift velocity. :) – Michael Brown Mar 30 '13 at 11:56
@MichaelBrown: Again No offense. But, you can try to answer via posts man. Your pre-commenting tradition makes me insert, "As Michael says..." everytime. I'm quite bad in typing ya know :D – Waffle's Crazy Peanut Mar 30 '13 at 12:02
Err.. Dear @LubošMotl: I think you've mistaken my answer. While $v_d$ is of the order of bare mm/s or cm/s, what I really mentioned is just the velocity of electron $v_e$ which can be upto a maximum $10^7$ m/s ;-) – Waffle's Crazy Peanut Mar 30 '13 at 12:21

The speed of the free electrons is fully determined by their density, cross section area, and current, see this numerical example, and it doesn't matter whether the current arises due to a battery or electromagnetic induction.

The drift velocity of electrons in common wires is very low, at most centimeters per second (it's less than a millimeter per second in the example above). This low speed doesn't prevent the signals (encoded in the electromagnetic fields, not in the location of electrons) from propagating nearly by the vacuum speed of light inside the wires; it's a completely different thing.

The estimate of the low drift velocity of the electrons is one of the favorite problems during oral physics qualifying exams, at least for Rutgers physics PhD students. ;-)

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