1
00:00:00,000 --> 00:00:00,640


2
00:00:00,640 --> 00:00:04,570
In the last video, we saw that
if we have two currents, or

3
00:00:04,570 --> 00:00:06,880
two wires carrying current, and
the current is going in

4
00:00:06,880 --> 00:00:09,160
the same direction, that they'll
attract each other.

5
00:00:09,160 --> 00:00:11,090
Now what would happen-- before
we break into the numbers--

6
00:00:11,090 --> 00:00:13,260
what would happen if the two
currents are going in opposite

7
00:00:13,260 --> 00:00:14,300
directions?

8
00:00:14,300 --> 00:00:15,980
Would they attract or
repel each other?

9
00:00:15,980 --> 00:00:17,460
And you can probably guess
that, but let's

10
00:00:17,460 --> 00:00:18,880
go through the exercise.

11
00:00:18,880 --> 00:00:21,240
Because I realize that last
time I did it, I got

12
00:00:21,240 --> 00:00:21,880
a little bit messy.

13
00:00:21,880 --> 00:00:23,430
And I'll do it a little bit
cleaner this-- I don't have to

14
00:00:23,430 --> 00:00:26,280
draw as many magnetic
field lines.

15
00:00:26,280 --> 00:00:29,390
So let's say that's wire 1.

16
00:00:29,390 --> 00:00:30,640
That's wire 2.

17
00:00:30,640 --> 00:00:34,130


18
00:00:34,130 --> 00:00:36,010
And I'll just make the currents
go in opposite

19
00:00:36,010 --> 00:00:36,500
directions.

20
00:00:36,500 --> 00:00:37,750
So this is I1.

21
00:00:37,750 --> 00:00:41,560


22
00:00:41,560 --> 00:00:43,420
And this is I2.

23
00:00:43,420 --> 00:00:45,390
So what would the magnetic
field created by

24
00:00:45,390 --> 00:00:46,500
current 1 look like?

25
00:00:46,500 --> 00:00:47,840
Well, let's do the
wrap around rule.

26
00:00:47,840 --> 00:00:51,090
Put our thumb in the direction
of the current, and then the

27
00:00:51,090 --> 00:00:52,870
magnetic field will
wrap around.

28
00:00:52,870 --> 00:00:57,200
It'll go into the page
here and it'll go

29
00:00:57,200 --> 00:00:58,720
out of the page here.

30
00:00:58,720 --> 00:01:00,240
If you put your thumb
up like that.

31
00:01:00,240 --> 00:01:02,410
Your right hand, always
use your right hand.

32
00:01:02,410 --> 00:01:03,940
And then you'll get that
type of magnetic field.

33
00:01:03,940 --> 00:01:06,890
And of course, it's going into
the page, into the video

34
00:01:06,890 --> 00:01:08,580
screen, all the way
out to infinity.

35
00:01:08,580 --> 00:01:10,170
It gets weaker and weaker.

36
00:01:10,170 --> 00:01:13,070
It's inversely proportional to
the radius away from the wire,

37
00:01:13,070 --> 00:01:13,960
so it'll get weaker
and weaker.

38
00:01:13,960 --> 00:01:18,160
But even here, this magnetic
field is going into the page.

39
00:01:18,160 --> 00:01:26,010
Now we know, just as a little
bit of review, the force

40
00:01:26,010 --> 00:01:29,740
created by current 1 on current
2-- that's just the

41
00:01:29,740 --> 00:01:31,530
convention I'm using, you
wouldn't always put the 1

42
00:01:31,530 --> 00:01:38,710
first-- is equal to current 2
times some length-- let's call

43
00:01:38,710 --> 00:01:40,910
that length 2-- along
the wire.

44
00:01:40,910 --> 00:01:43,340
This is going to be a vector
because it's a magnitude of

45
00:01:43,340 --> 00:01:45,000
length and a direction.

46
00:01:45,000 --> 00:01:50,560
And it goes in the same
direction as the current.

47
00:01:50,560 --> 00:01:53,260
So let's say that that is L2.

48
00:01:53,260 --> 00:01:55,450
So we're talking about
from here to here.

49
00:01:55,450 --> 00:01:57,510
That's the current.

50
00:01:57,510 --> 00:02:00,780
Cross product that with
the magnetic field.

51
00:02:00,780 --> 00:02:02,130
I'll switch back to that.

52
00:02:02,130 --> 00:02:05,730
The magnetic field
created by 1.

53
00:02:05,730 --> 00:02:07,500
Now it all seems pretty
complicated, but you can just

54
00:02:07,500 --> 00:02:11,400
take your right hand rule and
figure out the direction.

55
00:02:11,400 --> 00:02:14,350
So we put our index finger--
I'm doing it right now, you

56
00:02:14,350 --> 00:02:17,400
can't see it-- you put your
index finger in the

57
00:02:17,400 --> 00:02:23,990
direction of L2.

58
00:02:23,990 --> 00:02:25,280
You can write the 2 down
here, instead of

59
00:02:25,280 --> 00:02:26,920
writing a big 2 up there.

60
00:02:26,920 --> 00:02:30,500
Put your index finger in
the direction of L2.

61
00:02:30,500 --> 00:02:32,590
I keep redoing it just to make
sure I'm drawing it right.

62
00:02:32,590 --> 00:02:35,580


63
00:02:35,580 --> 00:02:38,720
Put your middle finger in the
direction of-- so this is L2.

64
00:02:38,720 --> 00:02:40,030
This is this.

65
00:02:40,030 --> 00:02:42,180
Goes in the direction of
the middle finger.

66
00:02:42,180 --> 00:02:43,230
Sorry, the index finger.

67
00:02:43,230 --> 00:02:44,480
Your middle finger is
going to go in the

68
00:02:44,480 --> 00:02:45,390
direction of the field.

69
00:02:45,390 --> 00:02:47,580
So it's going to be pointing
downwards, because the field

70
00:02:47,580 --> 00:02:51,830
is going into the page, on
this side of this wire.

71
00:02:51,830 --> 00:02:55,100
And then your other hands are
going to do what they will.

72
00:02:55,100 --> 00:02:56,520
And then your thumb is
going to go in the

73
00:02:56,520 --> 00:02:57,470
direction of the net force.

74
00:02:57,470 --> 00:03:00,710
So your thumb is going
to go like that.

75
00:03:00,710 --> 00:03:01,425
So there you have it.

76
00:03:01,425 --> 00:03:02,390
This is the top of your hand.

77
00:03:02,390 --> 00:03:04,560
You have your little veins or
tendons, whatever those are,

78
00:03:04,560 --> 00:03:06,850
that's your nail.

79
00:03:06,850 --> 00:03:09,590
So in this situation, when the
current is going in opposite

80
00:03:09,590 --> 00:03:13,470
direction, the net force is
actually going to be outward

81
00:03:13,470 --> 00:03:14,610
on this wire.

82
00:03:14,610 --> 00:03:16,170
The net force is outward.

83
00:03:16,170 --> 00:03:20,010
And then if you don't believe
me, you might want to try it

84
00:03:20,010 --> 00:03:25,200
yourself, but the force on
current 1 or on wire 1, or

85
00:03:25,200 --> 00:03:29,830
some length of wire 1, caused
by the magnetic field due to

86
00:03:29,830 --> 00:03:32,490
current 2, is also going
to be outwards.

87
00:03:32,490 --> 00:03:34,940
So here, if you want to think
about it little bit, or have a

88
00:03:34,940 --> 00:03:36,540
little bit of intuition, if
the current's going in the

89
00:03:36,540 --> 00:03:39,670
same direction they will
attract, and if currents are

90
00:03:39,670 --> 00:03:43,630
going in opposite directions
they will repel each other.

91
00:03:43,630 --> 00:03:45,510
So anyway, let's apply
some numbers.

92
00:03:45,510 --> 00:03:47,700
Let's apply some numbers
to this problem.

93
00:03:47,700 --> 00:03:51,670
Let's do it with the opposite
current direction.

94
00:03:51,670 --> 00:03:55,430
So let's say that current 1--
I'm just going to make up some

95
00:03:55,430 --> 00:03:58,130
numbers-- is 2 amperes.

96
00:03:58,130 --> 00:04:05,170
Current 2 is, I don't
know, 3 amperes.

97
00:04:05,170 --> 00:04:05,980
What else do you need to know?

98
00:04:05,980 --> 00:04:08,900
We need to know how far
apart they are.

99
00:04:08,900 --> 00:04:14,170
So let's say that this distance
right here is, I

100
00:04:14,170 --> 00:04:16,140
don't know, let's
say it's small.

101
00:04:16,140 --> 00:04:18,640
Let's try to get a respectable
number.

102
00:04:18,640 --> 00:04:23,420
Let's say that they're
1 millimeter apart.

103
00:04:23,420 --> 00:04:26,400
But we want everything in our
standard unit, so that all the

104
00:04:26,400 --> 00:04:27,100
units work out.

105
00:04:27,100 --> 00:04:28,070
So let's convert it to meters.

106
00:04:28,070 --> 00:04:33,390
So that equals 1 times 10
to the minus 3 meters.

107
00:04:33,390 --> 00:04:34,820
So they're pretty close apart.

108
00:04:34,820 --> 00:04:40,220
Now let's figure out the-- well,
let's do the force on

109
00:04:40,220 --> 00:04:42,260
wire 1 due to current 2.

110
00:04:42,260 --> 00:04:45,220
Just so that we can see that
this is also repelled.

111
00:04:45,220 --> 00:04:52,090
So let's say that the length in
question, L1, is equal to--

112
00:04:52,090 --> 00:04:57,170
I don't know, let's make it
a long wire-- 10 meters.

113
00:04:57,170 --> 00:04:57,630
All right.

114
00:04:57,630 --> 00:04:58,830
So how do we do this?

115
00:04:58,830 --> 00:05:01,710
So first let's figure out the
magnitude of the magnetic

116
00:05:01,710 --> 00:05:03,410
field created by I2.

117
00:05:03,410 --> 00:05:07,840


118
00:05:07,840 --> 00:05:10,150
I drew this hand too big,
took up too much space.

119
00:05:10,150 --> 00:05:15,260
So the magnetic field created
by current 2, worried about

120
00:05:15,260 --> 00:05:18,190
the magnitude of it,
that is equal--.

121
00:05:18,190 --> 00:05:19,560
And we saw before, we're
assuming that these

122
00:05:19,560 --> 00:05:21,040
are-- it's in air.

123
00:05:21,040 --> 00:05:24,190
So we can use the permeability
of a vacuum.

124
00:05:24,190 --> 00:05:25,720
So it's equal to that
constant, the

125
00:05:25,720 --> 00:05:27,630
permeability of a vacuum.

126
00:05:27,630 --> 00:05:30,150
Times I2.

127
00:05:30,150 --> 00:05:32,580
Just a magnitude.

128
00:05:32,580 --> 00:05:34,080
Now remember we figure out
the direction by wrapping

129
00:05:34,080 --> 00:05:34,900
our hand around it.

130
00:05:34,900 --> 00:05:35,900
We'll do that in a second.

131
00:05:35,900 --> 00:05:38,720
Divided by 2 pi times
the radius.

132
00:05:38,720 --> 00:05:41,980
So 2 pi radius.

133
00:05:41,980 --> 00:05:43,230
So let's see.

134
00:05:43,230 --> 00:05:45,390


135
00:05:45,390 --> 00:05:51,270
So the magnitude of the magnetic
field is equal to--

136
00:05:51,270 --> 00:05:55,560
well, we'll just keep that
that-- I2, I said is 3

137
00:05:55,560 --> 00:06:03,620
amperes-- times 3 amperes
divided by 2 pi times 1 times

138
00:06:03,620 --> 00:06:05,466
10 to the minus 3.

139
00:06:05,466 --> 00:06:10,220
And let's see, that answer
will be in teslas.

140
00:06:10,220 --> 00:06:11,300
All right.

141
00:06:11,300 --> 00:06:12,380
There, we already have the

142
00:06:12,380 --> 00:06:14,140
permeability of a vacuum there.

143
00:06:14,140 --> 00:06:15,090
So let's write that down.

144
00:06:15,090 --> 00:06:29,160
Permeability of a vacuum times 3
divided by 2 second pi times

145
00:06:29,160 --> 00:06:34,390
1E minus 3.

146
00:06:34,390 --> 00:06:39,490
And I get-- the answer will be
in teslas-- 6E minus 4 teslas.

147
00:06:39,490 --> 00:06:43,560
So the magnitude of the magnetic
field there created

148
00:06:43,560 --> 00:06:47,240
by current 2 is equal
to 6 times 10

149
00:06:47,240 --> 00:06:49,430
to the minus 4 teslas.

150
00:06:49,430 --> 00:06:51,170
Now what's the direction
of that magnetic field?

151
00:06:51,170 --> 00:06:52,640
So here we use our
wrap around rule.

152
00:06:52,640 --> 00:06:55,200
Take your right hand, wrap
it around the wire in the

153
00:06:55,200 --> 00:06:58,370
direction of the current, and
then you'll get the shape of

154
00:06:58,370 --> 00:06:59,260
the magnetic field.

155
00:06:59,260 --> 00:07:01,630
So I took my right hand,
my thumb goes in the

156
00:07:01,630 --> 00:07:02,940
shape of the current.

157
00:07:02,940 --> 00:07:06,420
My hand is going to look E like
this, and my knuckles.

158
00:07:06,420 --> 00:07:08,630
The fingers are going to
come out on that end.

159
00:07:08,630 --> 00:07:11,660
So the magnetic field caused by
current 2 is going to look

160
00:07:11,660 --> 00:07:13,000
something like that.

161
00:07:13,000 --> 00:07:15,970
So on this side of the wire,
where it intersects with the

162
00:07:15,970 --> 00:07:19,070
plane, it'll be popping out.

163
00:07:19,070 --> 00:07:22,050
And on this side, it'll
be popping in.

164
00:07:22,050 --> 00:07:27,840


165
00:07:27,840 --> 00:07:28,850
Fair enough.

166
00:07:28,850 --> 00:07:32,280
So now we can figure out
what the net force on

167
00:07:32,280 --> 00:07:34,120
this first wire is.

168
00:07:34,120 --> 00:07:36,390
Let me erase some of
this, just so I

169
00:07:36,390 --> 00:07:38,165
have some free space.

170
00:07:38,165 --> 00:07:40,870


171
00:07:40,870 --> 00:07:43,270
Let's see.

172
00:07:43,270 --> 00:07:46,410
We already used the 3 amperes,
we already used all of that.

173
00:07:46,410 --> 00:07:47,930
We already used all
of that, in fact.

174
00:07:47,930 --> 00:07:50,080
We just need to know that this
magnetic field, that's popping

175
00:07:50,080 --> 00:07:53,460
out of the page, we just need
to know its magnitude.

176
00:07:53,460 --> 00:07:55,940
We actually could even get rid
of this whole drawing, because

177
00:07:55,940 --> 00:07:58,030
now we just know that this has
created a magnetic field.

178
00:07:58,030 --> 00:07:59,160
And now we just worry
about the magnetic

179
00:07:59,160 --> 00:08:00,110
field and this wire.

180
00:08:00,110 --> 00:08:01,610
But anyway, I'll leave it there,
just so we remember

181
00:08:01,610 --> 00:08:03,560
what the whole problem was.

182
00:08:03,560 --> 00:08:06,750
So what's the net
force on wire 1?

183
00:08:06,750 --> 00:08:11,080
The net force on wire 1-- so we
could say caused by wire 2

184
00:08:11,080 --> 00:08:16,080
on wire 1, is equal to the
current in wire 1-- so that's

185
00:08:16,080 --> 00:08:21,270
2 amperes-- times the vector--
well, this is L1.

186
00:08:21,270 --> 00:08:24,880


187
00:08:24,880 --> 00:08:27,130
I'll write L1 right now.

188
00:08:27,130 --> 00:08:28,935
Cross the magnetic field.

189
00:08:28,935 --> 00:08:32,490


190
00:08:32,490 --> 00:08:33,909
And really, we just worry
about the magnitude.

191
00:08:33,909 --> 00:08:35,549
Because the direction, we can
figure out what the right--

192
00:08:35,549 --> 00:08:37,380
well, let's figure out the
direction, first of all.

193
00:08:37,380 --> 00:08:39,320
So L1 is going upwards.

194
00:08:39,320 --> 00:08:41,659
So that's the direction
of our index finger.

195
00:08:41,659 --> 00:08:43,520
B is going into the page.

196
00:08:43,520 --> 00:08:44,500
This is B2.

197
00:08:44,500 --> 00:08:46,460
That's the magnetic field
created by this wire.

198
00:08:46,460 --> 00:08:47,740
So it's going into the page.

199
00:08:47,740 --> 00:08:50,570
So if we use the right hand
rule, what happens?

200
00:08:50,570 --> 00:08:52,465
My index finger is going in the
direction of the current,

201
00:08:52,465 --> 00:08:54,320
in the direction of I1.

202
00:08:54,320 --> 00:08:56,600
My middle finger is pointed
downwards, so you can't see

203
00:08:56,600 --> 00:08:57,410
it's pointing into the page.

204
00:08:57,410 --> 00:08:59,350
And my other two fingers do
what they need to do.

205
00:08:59,350 --> 00:09:01,570
And so my thumb will
point in the

206
00:09:01,570 --> 00:09:03,510
direction of the net force.

207
00:09:03,510 --> 00:09:05,130
That's the top of my hand.

208
00:09:05,130 --> 00:09:08,560
So the net force is
in that direction.

209
00:09:08,560 --> 00:09:10,230
So we don't have to worry about
the vectors too much

210
00:09:10,230 --> 00:09:12,270
anymore, because
we know the end

211
00:09:12,270 --> 00:09:14,280
direction of the net force.

212
00:09:14,280 --> 00:09:16,270
So what's the magnitude?

213
00:09:16,270 --> 00:09:24,100
So the magnitude of the force
is equal to the current-- 2

214
00:09:24,100 --> 00:09:31,740
amperes-- times the magnitude
of the distance-- times 10

215
00:09:31,740 --> 00:09:36,140
meters-- times the magnitude
of the magnetic field.

216
00:09:36,140 --> 00:09:39,880
That's 6 times 10 to
the minus 4 teslas.

217
00:09:39,880 --> 00:09:41,530
And then when you take the cross
product, you take the

218
00:09:41,530 --> 00:09:44,090
sine of the theta between
these two vectors.

219
00:09:44,090 --> 00:09:44,920
But they're perpendicular.

220
00:09:44,920 --> 00:09:49,020
The magnetic field is going
into the page while the

221
00:09:49,020 --> 00:09:51,310
direction vector of the wire,
the length of the wire, is

222
00:09:51,310 --> 00:09:52,120
going along the page.

223
00:09:52,120 --> 00:09:52,895
So they are perpendicular.

224
00:09:52,895 --> 00:09:55,970
So the sine of theta just
comes out to be 1.

225
00:09:55,970 --> 00:09:57,120
So when things are perpendicular
you don't even

226
00:09:57,120 --> 00:09:58,310
have to worry about
that sine theta.

227
00:09:58,310 --> 00:10:01,060
You can almost just multiply
the terms and then use your

228
00:10:01,060 --> 00:10:02,860
right hand rule for
the direction.

229
00:10:02,860 --> 00:10:09,180
So anyway, this gives us 20
times 6-- 120 times 10

230
00:10:09,180 --> 00:10:10,560
to the minus 4.

231
00:10:10,560 --> 00:10:16,460
That's the same thing as 1.2
times 10 to the minus 2.

232
00:10:16,460 --> 00:10:21,020
So the magnitude of the force
is 1.2 force from

233
00:10:21,020 --> 00:10:23,640
current 2 on wire 1.

234
00:10:23,640 --> 00:10:27,340
The magnitude is 1.2 times 10
to the minus 2 Newtons.

235
00:10:27,340 --> 00:10:28,970
Because we used all
the right units.

236
00:10:28,970 --> 00:10:30,830
And the direction is outward.

237
00:10:30,830 --> 00:10:35,860
And so if we knew the mass of
this, we would-- you know, you

238
00:10:35,860 --> 00:10:38,000
just divide the force by the
mass, and you would know how

239
00:10:38,000 --> 00:10:40,860
fast it's accelerating at
that moment outwards.

240
00:10:40,860 --> 00:10:44,380
Of course, as it gets further
and further away, the magnetic

241
00:10:44,380 --> 00:10:46,630
field is going to get weaker,
so the net force is going to

242
00:10:46,630 --> 00:10:47,010
get weaker.

243
00:10:47,010 --> 00:10:50,150
So it'll start accelerating at
a slower and slower speed.

244
00:10:50,150 --> 00:10:50,410
Sorry.

245
00:10:50,410 --> 00:10:51,395
At a slower and slower rate.

246
00:10:51,395 --> 00:10:53,080
But of course, you're
still accelerating.

247
00:10:53,080 --> 00:10:55,580
So you are going to
continue to move

248
00:10:55,580 --> 00:10:57,330
away faster and faster.

249
00:10:57,330 --> 00:10:59,030
Anyway, all out of time.

250
00:10:59,030 --> 00:11:00,280
See you in the next video.

251
00:11:00,280 --> 00:00:00,000