1
00:00:00,000 --> 00:00:00,790


2
00:00:00,790 --> 00:00:05,240
So where I left off, we had
this infinite plate.

3
00:00:05,240 --> 00:00:07,900
It's just an infinite plane, and
it's a charged plate with

4
00:00:07,900 --> 00:00:09,410
a charge density sigma.

5
00:00:09,410 --> 00:00:12,830
And what we did is we said, OK,
well, we're taking this

6
00:00:12,830 --> 00:00:16,140
point up here that's h units
above the surface of our

7
00:00:16,140 --> 00:00:20,300
charge plate, and we wanted to
figure out the electric field

8
00:00:20,300 --> 00:00:25,150
at that point, generated by a
ring of radius r essentially

9
00:00:25,150 --> 00:00:31,400
centered at the base of where
that point is above.

10
00:00:31,400 --> 00:00:33,830
We want to figure out what is
the electric field generated

11
00:00:33,830 --> 00:00:36,130
by this ring at that point?

12
00:00:36,130 --> 00:00:42,450
And we figured out that the
electric field was this, and

13
00:00:42,450 --> 00:00:44,900
then because we made a symmetry
argument in the last

14
00:00:44,900 --> 00:00:47,110
video, we only care about
the y-component.

15
00:00:47,110 --> 00:00:49,700
Because we figured out that at
the electric field generated

16
00:00:49,700 --> 00:00:52,510
from any point, the x-components
cancel out,

17
00:00:52,510 --> 00:00:55,295
because if we have a point
here, it'll have some

18
00:00:55,295 --> 00:00:55,560
x-component.

19
00:00:55,560 --> 00:00:58,710
The field's x-component might
be in that direction to the

20
00:00:58,710 --> 00:01:00,650
right, but then you have another
point out here, and

21
00:01:00,650 --> 00:01:03,130
its x-component will
just cancel it out.

22
00:01:03,130 --> 00:01:05,760
So we only care about
the y-component.

23
00:01:05,760 --> 00:01:10,280
So at the end, we meticulously
calculated what the

24
00:01:10,280 --> 00:01:15,110
y-component of the electric
field generated by the ring

25
00:01:15,110 --> 00:01:18,570
is, at h units above
the surface.

26
00:01:18,570 --> 00:01:22,570
So with that out of the way,
let's see if we can sum up a

27
00:01:22,570 --> 00:01:26,480
bunch of rings going from radius
infinity to radius zero

28
00:01:26,480 --> 00:01:28,950
and figure out the total
y-component.

29
00:01:28,950 --> 00:01:30,750
Or essentially the total
electric field, because we

30
00:01:30,750 --> 00:01:33,490
realize that all the x's cancel
out anyway, the total

31
00:01:33,490 --> 00:01:37,120
electric field at that point,
h units above the

32
00:01:37,120 --> 00:01:39,930
surface of the plane.

33
00:01:39,930 --> 00:01:46,110
So let me erase a lot of this
just so I can free it up for

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00:01:46,110 --> 00:01:49,405
some hard-core math.

35
00:01:49,405 --> 00:01:54,130
And this is pretty much all
calculus at this point.

36
00:01:54,130 --> 00:01:55,640
So let me erase all of this.

37
00:01:55,640 --> 00:01:58,090
Watch the previous video if you
forgot how it was derived.

38
00:01:58,090 --> 00:02:01,370


39
00:02:01,370 --> 00:02:03,380
Let me even erase that because
I think I will

40
00:02:03,380 --> 00:02:04,630
need a lot of space.

41
00:02:04,630 --> 00:02:07,400


42
00:02:07,400 --> 00:02:09,220
There you go.

43
00:02:09,220 --> 00:02:12,770
OK, so let me redraw a little
bit just so we never forget

44
00:02:12,770 --> 00:02:17,390
what we're doing here because
that happens.

45
00:02:17,390 --> 00:02:20,390
So that's my plane that goes
off in every direction.

46
00:02:20,390 --> 00:02:25,110


47
00:02:25,110 --> 00:02:28,050
I have my point above the plane
where we're trying to

48
00:02:28,050 --> 00:02:29,130
figure out the electric field.

49
00:02:29,130 --> 00:02:30,740
And we've come to the conclusion
that the field is

50
00:02:30,740 --> 00:02:32,640
going to point upward, so
we only care about the

51
00:02:32,640 --> 00:02:33,860
y-component.

52
00:02:33,860 --> 00:02:39,590
It's h units above the surface,
and we're figuring

53
00:02:39,590 --> 00:02:43,200
out the electric field generated
by a ring around

54
00:02:43,200 --> 00:02:46,630
this point of radius r.

55
00:02:46,630 --> 00:02:49,010
And what's the y-component
of that electric field?

56
00:02:49,010 --> 00:02:50,760
We figured out it was this.

57
00:02:50,760 --> 00:02:53,230
So now what we're going to
do is take the integral.

58
00:02:53,230 --> 00:03:00,870
So the total electric field from
the plate is going to be

59
00:03:00,870 --> 00:03:06,440
the integral from-- that's a
really ugly-looking integral--

60
00:03:06,440 --> 00:03:09,620
a radius of zero to a
radius of infinity.

61
00:03:09,620 --> 00:03:11,970
So we're going to take a sum of
all of the rings, starting

62
00:03:11,970 --> 00:03:14,960
with a radius of zero all the
way to the ring that has a

63
00:03:14,960 --> 00:03:16,620
radius of infinity, because
it's an infinite plane so

64
00:03:16,620 --> 00:03:20,180
we're figuring out the impact
of the entire plane.

65
00:03:20,180 --> 00:03:24,060
So we're going to take the sum
of every ring, so the field

66
00:03:24,060 --> 00:03:26,240
generated by every ring, and
this is the field generated by

67
00:03:26,240 --> 00:03:28,275
each of the rings.

68
00:03:28,275 --> 00:03:29,710
Let me do it in a
different color.

69
00:03:29,710 --> 00:03:32,340
This light blue is getting
a little monotonous.

70
00:03:32,340 --> 00:03:47,050
Kh 2pi sigma r dr over
h squared plus r

71
00:03:47,050 --> 00:03:50,030
squared to the 3/2.

72
00:03:50,030 --> 00:03:51,280
Now, let's simplify
this a little bit.

73
00:03:51,280 --> 00:03:54,040
Let's take some constants out of
it just so this looks like

74
00:03:54,040 --> 00:03:56,410
a slightly simpler equation.

75
00:03:56,410 --> 00:04:04,795
So this equals the integral from
zero to-- So let's take

76
00:04:04,795 --> 00:04:06,670
the K-- I'm going to
leave the 2 there.

77
00:04:06,670 --> 00:04:08,370
You'll see why in a second, but
I'm going to take all the

78
00:04:08,370 --> 00:04:10,770
other constants out that we're
not integrating across.

79
00:04:10,770 --> 00:04:23,690
So it's equal to Kh pi sigma
times the integral from zero

80
00:04:23,690 --> 00:04:27,270
to infinity of what is this?

81
00:04:27,270 --> 00:04:28,930
So what did I leave in there?

82
00:04:28,930 --> 00:04:35,480
I left a 2r, so we could
rewrite this as-- well,

83
00:04:35,480 --> 00:04:37,560
actually, I'm running
out of space.

84
00:04:37,560 --> 00:04:50,470
2r dr over h squared plus r
squared to the 3/2, or we

85
00:04:50,470 --> 00:04:53,380
could think of it as the
negative 3/2, right?

86
00:04:53,380 --> 00:04:55,380
So what is the antiderivative
of here?

87
00:04:55,380 --> 00:04:58,800
Well, this is essentially the
reverse chain rule, right?

88
00:04:58,800 --> 00:05:00,850
I could make a substitution
here, if you're more

89
00:05:00,850 --> 00:05:03,390
comfortable using the
substitution rule, but you

90
00:05:03,390 --> 00:05:04,950
might be able to eyeball
this at this point.

91
00:05:04,950 --> 00:05:12,390
We could make the substitution
that u is equal to-- if we

92
00:05:12,390 --> 00:05:14,530
just want to figure out the
antiderivative of this-- if u

93
00:05:14,530 --> 00:05:18,300
is equal to h squared plus
r squared-- h is just a

94
00:05:18,300 --> 00:05:25,100
constant, right-- then du is
just equal to-- I mean, the du

95
00:05:25,100 --> 00:05:33,610
dr-- this is a constant, so it
equals 2r, or we could say du

96
00:05:33,610 --> 00:05:36,670
is equal to 2r dr.

97
00:05:36,670 --> 00:05:45,490
And so if we're trying to take
the antiderivative of 2r dr

98
00:05:45,490 --> 00:05:53,370
over h squared plus r squared to
the 3/2, this is the exact

99
00:05:53,370 --> 00:05:56,320
same thing as taking the
antiderivative with this

100
00:05:56,320 --> 00:05:57,310
substitution.

101
00:05:57,310 --> 00:05:59,680
2r dr, we just showed right
here, that's the same

102
00:05:59,680 --> 00:06:01,650
thing as du, right?

103
00:06:01,650 --> 00:06:06,100
So that's du over-- and then
this is just u, right?

104
00:06:06,100 --> 00:06:07,370
H squared plus r squared is u.

105
00:06:07,370 --> 00:06:08,590
We do that by definition.

106
00:06:08,590 --> 00:06:13,350
So u to the 3/2, which is equal
to the antiderivative

107
00:06:13,350 --> 00:06:20,260
of-- we could write this as
u to the minus 3/2 du.

108
00:06:20,260 --> 00:06:20,960
And now that's easy.

109
00:06:20,960 --> 00:06:25,100
This is just kind of reverse
the exponent rule.

110
00:06:25,100 --> 00:06:32,710
So that equals minus 2u to
the minus 1/2, and we

111
00:06:32,710 --> 00:06:33,530
can confirm, right?

112
00:06:33,530 --> 00:06:35,750
If we take the derivative of
this, minus 1/2 times minus 2

113
00:06:35,750 --> 00:06:40,320
is 1, and then subtract 1 from
here, we get minus 3/2.

114
00:06:40,320 --> 00:06:42,440
And then we could add plus c,
but since we're eventually

115
00:06:42,440 --> 00:06:43,590
going to do a definite
integral, the

116
00:06:43,590 --> 00:06:45,290
c's all cancel out.

117
00:06:45,290 --> 00:06:48,600
Or we could say that this is
equal to-- since we made that

118
00:06:48,600 --> 00:06:57,950
substitution-- minus 2 over--
minus 1/2, that's the same

119
00:06:57,950 --> 00:07:03,170
thing as over the square
root of h squared

120
00:07:03,170 --> 00:07:05,570
plus r squared, right?

121
00:07:05,570 --> 00:07:08,290
So all of the stuff I did in
magenta was just to figure out

122
00:07:08,290 --> 00:07:11,330
the antiderivative of this, and
we figured it out to be

123
00:07:11,330 --> 00:07:18,270
this: minus 2 over the
square root of h

124
00:07:18,270 --> 00:07:19,220
squared plus r squared.

125
00:07:19,220 --> 00:07:22,760
So with that out of the way,
let's continue evaluating our

126
00:07:22,760 --> 00:07:25,020
definite integral.

127
00:07:25,020 --> 00:07:29,170
So this expression simplifies
to-- this is a marathon

128
00:07:29,170 --> 00:07:33,920
problem, but satisfying-- K--
let's get all the constants--

129
00:07:33,920 --> 00:07:43,320
Kh pi sigma-- we can even take
this minus 2 out-- times minus

130
00:07:43,320 --> 00:07:46,970
2, and all of that, and we're
going to evaluate the definite

131
00:07:46,970 --> 00:07:53,910
integral at the two boundaries--
1 over the square

132
00:07:53,910 --> 00:08:00,700
root of h squared plus r squared
evaluated at infinity

133
00:08:00,700 --> 00:08:05,040
minus it evaluated
at 0, right?

134
00:08:05,040 --> 00:08:09,150
Well, what does this
expression equal?

135
00:08:09,150 --> 00:08:11,840
What is 1 over the square
root of h squared

136
00:08:11,840 --> 00:08:13,280
plus infinity, right?

137
00:08:13,280 --> 00:08:15,210
What happens when we evaluate
r at infinity?

138
00:08:15,210 --> 00:08:18,420
Well, the square root of
infinity is still infinity,

139
00:08:18,420 --> 00:08:21,955
and 1 over infinity is 0, so
this expression right here

140
00:08:21,955 --> 00:08:23,140
just becomes 0.

141
00:08:23,140 --> 00:08:27,500
When you evaluate it at
infinity, this becomes 0 minus

142
00:08:27,500 --> 00:08:29,750
this expression evaluated
at 0.

143
00:08:29,750 --> 00:08:31,670
So what happens when
it's at 0?

144
00:08:31,670 --> 00:08:35,710
When r squared is 0, we get 1
over the square root of h

145
00:08:35,710 --> 00:08:37,960
squared, right?

146
00:08:37,960 --> 00:08:39,080
So let's write it all out.

147
00:08:39,080 --> 00:08:47,730
This becomes minus 2Kh pi sigma
times 0 minus 1 over the

148
00:08:47,730 --> 00:08:49,140
square root of h squared.

149
00:08:49,140 --> 00:09:01,870
Well this equals minus 2Kh pi
sigma times-- well, 1 over the

150
00:09:01,870 --> 00:09:05,080
square root of h squared, that's
just 1 over h, right?

151
00:09:05,080 --> 00:09:09,130
And there's a minus times
minus 1 over h.

152
00:09:09,130 --> 00:09:12,820
Well, this minus and that
minus cancel out.

153
00:09:12,820 --> 00:09:17,730
And then this h and this 1
over h should cancel out.

154
00:09:17,730 --> 00:09:21,170
And all we're left with, after
doing all of that work, and

155
00:09:21,170 --> 00:09:23,590
I'll do it in a bright color
because we've done a lot of

156
00:09:23,590 --> 00:09:28,760
work to get here,
is 2K pi sigma.

157
00:09:28,760 --> 00:09:31,380


158
00:09:31,380 --> 00:09:33,150
So let's see it at
a lot of levels.

159
00:09:33,150 --> 00:09:35,270
First of all, what did
we even do here?

160
00:09:35,270 --> 00:09:37,010
We might have gotten
lost in the math.

161
00:09:37,010 --> 00:09:41,640
This is the net electric, the
total electric field, at a

162
00:09:41,640 --> 00:09:48,970
point at height h above this
infinite plate that has a

163
00:09:48,970 --> 00:09:52,390
uniform charge, and the charge
density is sigma.

164
00:09:52,390 --> 00:09:55,250
But notice, this is the electric
field at that point,

165
00:09:55,250 --> 00:09:56,700
but there's no h in here.

166
00:09:56,700 --> 00:10:00,430
So it essentially is telling
us that the strength of the

167
00:10:00,430 --> 00:10:05,320
field is in no way dependent on
how high above the field we

168
00:10:05,320 --> 00:10:08,160
are, which tells us this is
going to be a constant field.

169
00:10:08,160 --> 00:10:10,020
We can be anywhere above
the plate and the

170
00:10:10,020 --> 00:10:11,540
charge will be the same.

171
00:10:11,540 --> 00:10:13,830
The only thing-- oh, sorry,
not the charge.

172
00:10:13,830 --> 00:10:18,600
The field will be the same, and
if we have a test charge,

173
00:10:18,600 --> 00:10:20,110
the force would be the same.

174
00:10:20,110 --> 00:10:23,000
And the only thing that the
strength of the field or the

175
00:10:23,000 --> 00:10:25,450
strength of the exerted
electrostatic force is

176
00:10:25,450 --> 00:10:28,090
dependent on, is the charge
density, right?

177
00:10:28,090 --> 00:10:32,130
This is Coulomb's constant, pi
is pi, 2pi, and I think it's

178
00:10:32,130 --> 00:10:34,100
kind of cool that it involves
pi, but that's

179
00:10:34,100 --> 00:10:35,465
something else to ponder.

180
00:10:35,465 --> 00:10:39,640
But all that matters is
the charge density.

181
00:10:39,640 --> 00:10:43,210
So hopefully, you found that
reasonably satisfying, and the

182
00:10:43,210 --> 00:10:46,930
big thing that we learned here
is that if I have an infinite

183
00:10:46,930 --> 00:10:52,500
uniformly charged plate, the
field-- and I'm some distance

184
00:10:52,500 --> 00:10:57,860
h above that field-- above that
plate, it doesn't matter

185
00:10:57,860 --> 00:10:58,750
what that h is.

186
00:10:58,750 --> 00:11:02,210
I could be here, I could be
here, I could be here.

187
00:11:02,210 --> 00:11:05,560
At all of those points, the
field has the exact same

188
00:11:05,560 --> 00:11:08,300
strength, or the net
electrostatic force on a test

189
00:11:08,300 --> 00:11:11,470
charge at those points has the
exact same strength, and

190
00:11:11,470 --> 00:11:13,290
that's kind of a neat thing.

191
00:11:13,290 --> 00:11:15,590
And now if you do believe
everything that occurred in

192
00:11:15,590 --> 00:11:20,980
the last two videos, you can
now believe that there are

193
00:11:20,980 --> 00:11:23,400
such things as uniform electric
fields and they occur

194
00:11:23,400 --> 00:11:25,670
between parallel plates,
especially far away from the

195
00:11:25,670 --> 00:11:26,770
boundaries.

196
00:11:26,770 --> 00:00:00,000
See you soon.