00:01
In this problem, you're given potential energy function of the electron liquid helium surface, and x being the distance from that the electron is from the surface, a positive quantum.
00:16
And they ask you some questions about this function and what it can give us.
00:19
First one is what is the force at any particular x? so f of x is equal to minus d u d x.
00:28
The minus sign is that we want the force pointing the direction of decreasing potential energy, not increasing.
00:38
That's where the minus sign comes in.
00:40
So think of gravity.
00:42
Potential energy increases as you go in height, but the force is downward.
00:47
If you go in that direction, you're decreasing, potential energy decreasing height.
00:54
So let's do the derivative here.
00:56
D -dx minus a, x minus 2, plus b.
01:02
X minus 4 and it's because minus minus a minus 2 x minus 3 plus b minus 4 x minus 5 and now we can write that out we got minus times a minus times a minus so that's a plus it's going to be a minus so minus 2a or x cubed minus times a minus plus plus 4 b or or x to the fifth and that's what they want you to show so we have the force now at any point the next part wants to know where is the place where the electron would be trapped and let's look at this let's look at this graphically first though do d x is a slope of a tangent line at the graph any point you want you want to know what's going on here draw a tangent line they're and calculate the slope.
02:03
F is minus of that value.
02:06
Now, in this region here, all the slopes are negative.
02:11
So that means the force is positive.
02:14
In this region here, all the slopes are positive.
02:19
And so that means the force is negative.
02:24
Now here we have the one point where the tangent line is horizontal.
02:28
That means the force is zero.
02:36
So here where the force is zero, as i said, f here is going to be less than zero, f here is great in zero.
02:45
So if you were to put electron here, say, he's going to feel a force that wants to take him to larger x.
02:53
So he's certainly not trapped there.
02:55
Try to put him, well, you know, i'm using, i'm using this as a guy, but i remember it's all along one direction.
03:01
If i try to put him effectively here, then he's feeling a force that wants to take him to lesser x.
03:09
So no one's state.
03:10
Anywhere.
03:13
So those are not equilibrium trap points.
03:15
If we put him here, though, the force is zero.
03:18
He's got nothing driving him any which way.
03:21
So just place him there, he's going to stay there.
03:23
Now, beyond that, if you wanted to look at what type of equilibrium it is, if you made an infinitesimal motion, tried to get away, but in an infinitesimal manner, say to lesser x, this force would say, sorry, you're not going anywhere.
03:41
You're staying where you are.
03:43
Likewise, try to go to higher, higher x.
03:45
This force would tell you that, no, you're not going anywhere either.
03:49
If it tells me a motion, you're really staying at the same spot, and you're not doing a finite motion.
03:54
If you did a finite motion, think of this as your mechanical energy line here, this horizontal line.
04:00
E equals you, you are turning points.
04:03
You'd have two turning points.
04:04
You'd be going back and forth between those two turning points.
04:07
Remember turning points? before it, you're going one direction.
04:10
You connect energy zero at that point, turning point, then you go back the other way you came...