What is the relationship between displacement, speed, and time in a random walk?

[Andy_Taiwanese]

http://imgur.com/a/hFubT

http://imgur.com/a/fquuF

In the first image, first line in the last paragraph.

If carriers move with an average velocity component Vd in the x direction(along the wire), the displacement they experience in this direction in a time interval delta t is delta x = Vd * delta t. The speed Vd of the charge carrier along the wire is an average speed called the drift speed.

The first sentence define that carriers move with an average velocity component Vd in the x direction(in fact, carriers also move with an average velocity component in the y direction). The second sentence states that the "speed" Vd of the charge carrier along the wire is an average speed called the drift speed(speed is the total distance divided by time, that is, it also include the average component in the y direction).

I may want to ask that is my interpretation correct?

 

[Andy_Taiwanese]

View attachment 207495View attachment 207496

This is my first post, could you teach me how to display image on the post?

 

[davenn]

The first sentence define that carriers move with an average velocity component Vd in the x direction(in fact, carriers also move with an average velocity component in the y direction). The second sentence states that the "speed" Vd of the charge carrier along the wire is an average speed called the drift speed(speed is the total distance divided by time, that is, it also include the average component in the y direction).

what are you defining as the y direction ?

I may want to ask that is my interpretation correct?

I don't think so

 

[davenn]

Is this homework ?

 

[Andy_Taiwanese]

Is this homework ?

No, that's not my homework, I am the physics hobbyist

 

[Andy_Taiwanese]

The image uploaded on the website is resolution decreased, is there other way to upload?

 

[davenn]

No, that's not my homework, I am the physics hobbyist

OK

so please answer my earlier question ... what are you defining as the y direction ?

 

[Andy_Taiwanese]

OK

so please answer my earlier question ... what are you defining as the y direction ?

vd is the average velocity in x direction. There must be the average velocity in y direction, let's say vy.

 

[davenn]

There must be the average velocity in y direction, let's say vy.

that didnt really answer my Q

current is along the length of the wire ie. the x direction for DC it is one direction only
for AC it alternates back and forward ... how could it be in any other direction ?

 

[Andy_Taiwanese]

that didnt really answer my Q

current is along the length of the wire ie. the x direction for DC it is one direction only
for AC it alternates back and forward

Sorry, I am in bad English...

 

[davenn]

Sorry, I am in bad English...

that's OK

did you understand my reply ?

 

[Andy_Taiwanese]

that's OK

did you understand my reply ?

Although being along with a wire, the electron is actually moving like zigzag motion.

 

[davenn]

Although being along with a wire, the electron is actually moving like zigzag motion.

yes that is correct. BUT that isn't the current flow that is just the standard random motion of the electrons

When a voltage potential is applied to the wire ( the electrical circuit), that random movement doesn't stop
rather now the electric field generated causes the electrons to also have a overall movement along the length of the wire
and THAT is the drift and the current

Dave

 

[Andy_Taiwanese]

yes that is correct. BUT that isn't the current flow that is just the standard random motion of the electrons

When a voltage potential is applied to the wire ( the electrical circuit), that random movement doesn't stop
rather now the electric field generated causes the electrons to also have a overall movement along the length of the wire
and THAT is the drift and the current

Dave

Thank you! I figured it out!
May I ask the other questions further?

 

[davenn]

Thank you! I figured it out!
May I ask the other questions further?

awesome

just start a new thread for a new topic
and best to use the B for Basic tag

 

[Andy_Taiwanese]

http://imgur.com/a/UXMx3
In middle paragraph this image."In our structural model, we shall assume that the excess kinetic energy acquired by the electrons in the electric field is lost to conductor in the collision process.

What is words meaning with red color? as far as conductor is concerned?

For whole sentence, that is, for those electrons in the collision process whose kinetic energy offered by electric field is dissipating and converting to the heat. right?

 

[Andy_Taiwanese]

awesome

just start a new thread for a new topic
and best to use the B for Basic tag

Actually it's the same topic, should I start with a new thread?

 

[davenn]

Actually it's the same topic, should I start with a new thread?

keep it here then :)

 

[Andy_Taiwanese]

keep it here then :)

Thank you, I already posted the question :)

 

[sophiecentaur]

There must be the average velocity in y direction, let's say vy.

The Average (i.e. the mean) velocity in the y direction will be zero. The RMS speed in the y direction could be anything, depending on the temperature of the wire because the average Kinetic Energy of electrons corresponds to the temperature.

excess kinetic energy acquired by the electrons in the electric field is lost to conductor in the collision process.

What is words meaning with red color? as far as conductor is concerned?

The interaction of the free electrons with the lattice of the metal will cause the wire to heat up and the Electrical Energy Lost per Coulomb of charge is the Potential Drop (Volts) across the length of wire. (Joules Per Coulomb is Volts).

 

[Andy_Taiwanese]

The Average (i.e. the mean) velocity in the y direction will be zero. The RMS speed in the y direction could be anything, depending on the temperature of the wire because the average Kinetic Energy of electrons corresponds to the temperature.

The interaction of the free electrons with the lattice of the metal will cause the wire to heat up and the Electrical Energy Lost per Coulomb of charge is the Potential Drop (Volts) across the length of wire. (Joules Per Coulomb is Volts).

When voltage is applied, the electron would be forced to move, the moving electron itself is with kinetic energy. Only with moving electron, there is current exists.
I would like to ask what is "excess kinetic energy"? It should have kinetic energy.

 

[Andy_Taiwanese]

http://imgur.com/a/UXMx3
At the last paragraph, "the motion of the electron through the metal is characterized by a very large number of collisions per second. Consequently, we consider the average value of v bar over a time interval long compared with the time interval between collisions, which gives us the drift velocity vd bar

I don't know what this sentence means. Could anyone please guide me?

 

[jbriggs444]

I don't know what this sentence means. Could anyone please guide me?

This will be a long winded explanation... Maybe it will be worth something.

Suppose that you are drunk. You walk out the front door of the bar (located at a corner), randomly pick a direction and start walking. Every time you get to another corner, you randomly pick a new direction (left, right, forward or back) and keep walking some more. This is a "drunkard's walk" in two dimensions.

If you stagger at an speed of 1 meter per second then after walking your first block your average velocity will be 1 meter per second. After walking 2 blocks, your average velocity will be somewhat less. That's because you might have spent the second block walking in a different direction. [Recall that your average velocity is your total displacement (0 blocks, $\sqrt{2}$ blocks or 2 blocks) divided by the elapsed time (2 blocks worth)].

After walking a third block, your average velocity will be even less. It turns out that for a large variety of random walks, the average distance of the final point from the initial point increases as the square root of the number of steps.

That means that the average quotient: $\frac{displacement}{time}$ scales as with the number of moves n as $\frac{\sqrt{n}}{n} = \frac{1}{\sqrt{n}}$

That is to say that a drunk's average velocity tends to zero as the number of blocks he covers becomes larger and larger.

If we cast an electron in the role of a drunk, the displacement from one bounce to the next as the moves on his walk and the total number of moves as how long you wait before taking the average then... the average velocity of a thermally bouncing electron tends to zero over the long term.

If you add an electric field, this is no longer true. The electron's "drunk walk" is still random. But it is biased in a particular direction. If you average over the long term, you no longer get a result of zero. You get a result that reflects that bias.

Edit: If you work that square root thing backwards and apply to a single electron, "long term" needs to be on the order of the time for one bounce times $(\frac{thermal\ velocity}{drift\ velocity})^2$

 

[Andy_Taiwanese]

This will be a long winded explanation... Maybe it will be worth something.

After walking a third block, your average velocity will be even less. It turns out that for a large variety of random walks, the average distance of the final point from the initial point increases as the square root of the number of steps.

That means that the average quotient: $\frac{displacement}{time}$ scales as with the number of moves n as $\frac{\sqrt{n}}{n} = \frac{1}{\sqrt{n}}$

If we cast an electron in the role of a drunk, the displacement from one bounce to the next as the moves on his walk and the total number of moves as how long you wait before taking the average then... the average velocity of a thermally bouncing electron tends to zero over the long term.

If you add an electric field, this is no longer true. The electron's "drunk walk" is still random. But it is biased in a particular direction. If you average over the long term, you no longer get a result of zero. You get a result that reflects that bias.

Edit: If you work that square root thing backwards and apply to a single electron, "long term" needs to be on the order of the time for one bounce times $(\frac{thermal\ velocity}{drift\ velocity})^2$

"After walking a third block, your average velocity will be even less" I know this, because the total displacement is constant but time is more long.
"It turns out that for a large variety of random walks, the average distance of the final point from the initial point increases as the square root of the number of steps."
I know what is distance but I don't know what is average distance and why average distance would increase as the square root of the number of steps?
Also, I know what is quotient but I don't know what is average quotient.
"If we cast an electron in the role of a drunk, the displacement from one bounce to the next as the moves on his walk and the total number of moves as how long you wait before taking the average then... the average velocity of a thermally bouncing electron tends to zero over the long term."
Sorry...my English is poor. I don't understand this sentence.

 

[jbriggs444]

"After walking a third block, your average velocity will be even less" I know this, because the total displacement is constant but time is more long.

After zero moves, the average distance of drunk from bar is 0 blocks.
After one move, the average distance of drunk from bar is 1 block.
After two moves, the average distance of drunk from bar is $1 + \frac{\sqrt{2}}{2}$ ~= 1.7 blocks.
After three moves, the average distance of drunk from bar is about 1.96 blocks if my calculations are correct.

"It turns out that for a large variety of random walks, the average distance of the final point from the initial point increases as the square root of the number of steps."
I know what is distance but I don't know what is average distance and why average distance would increase as the square root of the number of steps?

It is a principle of statistics. You can trace it back to facts like "the variance of a sum (of random variables) is equal to the sum of the variances". The variance measures squared deviation from the mean. If you take the square root of the variance you get a number that is roughly (but not exactly) the same as the average distance from the mean.

The mean displacement (a vector quantity) of the drunk after any number of steps is clearly zero. If you average his position among all possible paths, that average is exactly at the door to the bar.

The mean distance (a scalar quantity) of the drunk from the bar can be roughly estimated as the square root of the variance (alternately, this is the same thing as "standard deviation"). The variance is proportional to the number of steps. The standard deviation is proportional to the square root of the number of steps.

Also, I know what is quotient but I don't know what is average quotient.

If you compute the quotient separately for every possible path and then take the average of those quotients, that is the result I am describing.

"If we cast an electron in the role of a drunk, the displacement from one bounce to the next as the moves on his walk and the total number of moves as how long you wait before taking the average then... the average velocity of a thermally bouncing electron tends to zero over the long term."
Sorry...my English is poor. I don't understand this sentence.

An electron bounces in three dimensions at arbitrary angles. A drunk bounces in two dimensions along a grid. Those differences are irrelevant. Both travel according to a random walk. Both will have an average distance moved that is roughly proportional to the square root of how long you watch and an average $\frac{total\ distance}{total\ time}$ that is is roughly inversely proportional to the square root of how long you watch.