00:02
This problem seeks to answer the question of do cell phones generate dangerous magnetic fields? and it actually gets a little bit complicated, so i'm going to give you a shortcut in case you just want the short answer.
00:17
But i will go through the actual derivation of everything.
00:20
The first question is very easy, so we don't have to skip anything for that.
00:23
The first part of this question says, if the cell phone battery has a potential of 1 .7 volts, and it generates a power of 1 .5 watts, what do you estimate the current generated by the battery to be? so here we're just going to use a very sort of simple -minded expression here and just say, well, the power generated by a current going across the potential is equal to e, i, the current times the potential.
00:54
And so if we solve that for the current, we get that the current is equal to the power, divided by the potential, and that's equal to 1 .5 watts divided by 1 .7 volts.
01:08
And remember, we're using all si units here, so the answer is going to come out in ampers.
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So this is going to be 0 .8824 amper.
01:20
And so that's what the current generated by the battery is.
01:26
Now, we're going to go now with this is that, we're going to take that current and we're going to use that current to have that current generate a magnetic field.
01:39
And so we're going to give you the sort of basic parameters of the system.
01:43
We're going to assume that the diameter of your phone speaker is 3 centimeters and that the width of your head is about 20 centimeters.
01:51
And these are roughly actually about true.
01:53
And then we're going to assume we treat the current in the phone speaker as a current loop with the same diameter as the speakers.
02:01
We're going to treat this speaker as if it's a current loop, the diameter of three centimeters.
02:08
And then assuming those values, we're going to assume that the speaker current is equal to that battery current that we just calculated.
02:18
We're going to use that to estimate the magnetic field generated by the phone right in the center of your head in between your two ears.
02:26
Presuming the center of your head is right in between your two ears.
02:30
Most people, that's true.
02:31
Okay, so now if you want to skip all of this and go to the end, i'm going to give you the expression for the magnetic field in terms of all these parameters.
02:43
And that's just equal to magnetic field is equal to mu not.
02:47
That's the magnetic permeability constant of four pi times 10 to minus seven tesla meters for amplier.
02:55
And i is that current that we just calculated.
02:58
R is the radius of the speaker loop.
03:00
So that's 1 .5 centimeters.
03:02
You have to convert that to meters.
03:05
And z here is the distance halfway in between your head.
03:09
So that's 10 centimeters in this case.
03:11
If you want to just skip to that, you can do that.
03:14
But i'm going to go through and derive this expression so you see where this comes from.
03:18
Now the picture we have of this whole system is that i've got this current loop which is in red.
03:25
That's the loop going around the speaker.
03:30
And then this line, this green line that goes up labeled z that's the distance from the center of that loop up into the center of your head so the center of your head is at this point p that's the field point that's where we're going to calculate the magnetic field and the radius of that loop is is r which is uh 1 .5 centimeters and that height z is actually equal to 10 centimeters this picture is not to scale uh i don't there's no value in doing it to scale and it just makes it hard r to c stuff, so i'm just going to leave it as it is.
04:05
We're going to use what's called the biosavart law.
04:08
And the biosavart law, this tells us that the differential magnetic field due to a bit of current moving into differential length of line.
04:18
So the current is i, and the differential length of wire is ds.
04:24
The relationship, that magnetic field is equal to mu not times that current, times that different, differential lengths crossed into r, and r is the vector that goes from that differential bit of current to the field point.
04:41
So it's a cross product there.
04:44
You remember that from earlier in your physics 201 adventure.
04:49
And then divided by four pi times the cube of this distance from that current element, that differential element of current to the field point.
04:58
So that's the differential element of current.
05:00
We're ultimately going to have to integrate that around the current loop.
05:04
We'll get to that in a minute.
05:06
I just want to show you first of all the geometry of this because it results in some simplifications which are actually really valuable.
05:15
So here if i take this little bit of current, this ds that's moving in this little loop and then i have r and r is this vector that goes from that differential element of current up to the field points.
05:30
So if i was going to draw on the field point in here, i would say the field point is right, where is it? field point is right up there.
05:41
Okay, so then if i put in the, if i take the cross product of ds and r, the direction that's gonna point in is shown here by this direction of this differential element of magnetic field.
05:58
In other words, if ds points this way, and r points this way, then, i'll try it this way.
06:06
If ds points this way and r points this way, then the differential magnetic field points this way.
06:16
Now, the magnitude we're gonna have to deal with in a minute, but that's the basic directions involved.
06:24
So let's go on and see what happens next.
06:27
Now, so here's the picture of all of these things together.
06:31
There's a differential element of current.
06:34
And there's this z distance here up from the center of the loop into the field point.
06:41
And r is that radius of that circle.
06:44
And here's this differential element of magnetic field.
06:48
Now, what i'm going to do is i'm going to integrate this around the entire circle.
06:53
And as i do that, the direction of ds changes.
06:57
And so what i'm going to end up happening is i'm going to have this series of all of these different contributions of the differential magnetic field.
07:08
So all these different guys are trying and draw this in here.
07:11
So i got one there and i got one there over here and i got one there over here.
07:18
You know, all these ones, they all come from different parts of the differential element of current as i go around that circle.
07:27
And i'm going to add those all up.
07:28
That's what integrating is.
07:29
So i'm going to add all those up and what you see then is that the horizontal components vanish.
07:37
Okay, so the horizontal components vanish and the vertical components add...