00:01
In this problem, we have a parallel plate capacitor.
00:03
We have a uniform, which gives us a uniform electric field inside between the plates.
00:09
Notice the even spacing, somewhat even spacing, or by electric field lines.
00:14
And they're going to ask us some questions about electric potential, electric potential energy, kinetic energy.
00:21
So if there's problems, it will take one at a time.
00:23
First thing it wants is to talk about, find the formula for the potential.
00:29
And we're going to, our potential is going to be zero.
00:32
At the bottom plate.
00:35
It tells us to do that.
00:36
So they want the potential at any particular h value.
00:44
Now before we do that though, i'm gonna talk in general.
00:47
Here is a point a, here is a point b.
00:51
Two different h values, and i'm gonna go on a path between a straight from a to b.
01:05
I could go on a zigzag path and i'll talk about that in a second, not just yet.
01:08
But my infinitesimo steps will be just straight up.
01:14
You'd be doing infinite collection of dl vectors, and then in the end, your overall displacement vectors will be b.
01:22
Add them all up and you get to b.
01:26
So this is minus point a to point b.
01:31
E .d .l.
01:35
I should mention some books you use ds here.
01:38
It doesn't matter.
01:39
Same thing.
01:41
What is the meaning behind this? it's the work you must do to move a unit positive charge from a to b without changing its kinetic energy.
01:52
That's its meaning.
01:57
To move a unit positive charge from a to b without changes kinetic energy.
02:03
Now, you might say, let's talk about now the integral.
02:06
Does it, did it matter if you go to a to b that way? what is if you take a zigzag path or a rectangular path of some type? does it matter? electrostatic field is a conservative field.
02:18
Field, the work is independent of path.
02:25
So going from a to b this way or any other path is going to give me the same exact result.
02:32
And notice, has to.
02:34
I could not introduce this because this only depends on where i began and where i end.
02:39
If this depended on the path, then i can't write this.
02:42
I cannot define a potential at all, just like i couldn't define a potential energy when you did gravitational work in mechanics.
02:54
The work must be path independent to be able to define a potential energy or in this case potential.
03:01
It has to be.
03:02
So you always choose a nice simple path.
03:04
You have to go from one to the other.
03:07
So that's because conservative field.
03:17
So on our case, a is going to be zero and b is going to be h.
03:22
Now, i'm going to introduce, i'm not going to use the h as a a variable, i'm just going to use it as really a y value.
03:33
Y is more important to me than h.
03:36
That just identifies.
03:37
To me, it's just a ruler measurement.
03:40
Y, though, has a sign to it.
03:43
And you'll understand y in a minute, why i do it.
03:46
So there's my axis.
03:47
So the bottom plate is y equals zero.
03:50
So we're fine.
03:52
Okay.
03:55
V of h, v of zero, minus from zero to h.
04:05
Now, the dl's vectors is going to have a component, d .y, j, hat.
04:16
Now, d .y is a component.
04:18
That means it could be positive, could be negative.
04:21
What tells you what it will be, is your sense of your, you've noticed my limits of integration are going from zero to h, an increase.
04:29
So it means my dy's, my components along any step, positive, my infinitesimal steps are positive.
04:37
In this case.
04:38
If i had gone from h to 0, my dy's in that case would be negative.
04:44
Dy's component, component, component of the dl vector.
04:53
It's not a magnitude.
04:54
I'll teach you the second.
04:55
I'll show you how to do this from the magnitude standpoint.
04:58
But not yet.
04:59
We're doing the full vector standpoint right now.
05:03
So this is minus 0 to h.
05:07
Now e.
05:08
Dot j hat.
05:09
Anytime you dot a vector with a unit vector, you get the component in that direction.
05:13
So this becomes ey, dy.
05:21
But ey, it's in the negative y direction.
05:24
That's minus e, dy.
05:29
And e is constant, so i can bring that out.
05:31
This becomes e, 0 to h, dy, and that's a trivial integration, h, eh.
05:43
And they told us v at zero is zero, so we'll get rid of that.
05:50
So our final answer, v of h is equal to e, times h.
05:55
There is our answer.
05:57
Notice the potential rises with h.
05:59
Remember that for what i'm going to tell you in a few minutes.
06:03
So that's how you do it.
06:05
That's how you do it.
06:07
Again, this would be the amount of work you must do to bring a unit positive charge from the bottom plate to position h.
06:16
Without changes connect energy.
06:17
That's the meaning.
06:19
Now let me give a slight digression here.
06:22
I'll take you through this.
06:23
If you're not comfortable with this vector work, let me show you what it would mean to do it with the other form of the dot product.
06:34
So v of h, v .0 again, minus.
06:40
Now this will be e, oh, i put in my zero to h, e, dl, which is now a length.
06:47
It's got no sign to it.
06:49
It is the magnitude of the vector.
06:52
There's no minus sign anymore.
06:55
So that means these guys, you must go from zero to h.
06:59
You must increase.
07:00
Your infinitesimals must be.
07:01
Positive.
07:02
Here they could be positive or negative depending on which way you are going.
07:07
Here must be positive, must be, because this is a magnitude.
07:12
Coside, the vectors are in opposite directions.
07:16
E is minus y, dl vector is positive y, so 180, which is minus 1.
07:22
So, 180, which is minus 1.
07:23
So this becomes integral of 0 to h, e, dl, which is e, dl, which just gives me the length...