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### Author Topic: Does magnetic field strength follow an inverse square law?  (Read 35296 times)

#### chris

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##### Does magnetic field strength follow an inverse square law?
« on: 19/03/2010 10:13:39 »
I thought I'd post this as there seems to be quite a bit of confusion on the net about this issue.

Put simply, does the strength of a magnetic field, measured at increasing distance from the field source, decay according to 1/r^2

Chris

#### JP

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##### Does magnetic field strength follow an inverse square law?
« Reply #1 on: 19/03/2010 11:53:18 »
The 1/r^2 law results from what are called monopoles, which are basically point-sources of the field.  Single electric charges are monopoles which is why the field of a single electric source goes like 1/r^2 as you move away from it.  If you put a positive and negative charge near each other (and keep them separated), you have what's called a dipole.  Since there's one positive and one negative charge, there's a 1/r^2 field pointing in towards the charges and a 1/r^2 field pointing away from the charges.  The 1/r^2 terms cancel each other out and you end up with the next biggest term, which is 1/r^3.  This is called an electric dipole field.  You can have quadrupoles, octopoles, etc., each one constructed by canceling out the previous -pole which leads to a faster decay in powers of 1/r.

Magnets are a little trickier since no one has ever seen a magnetic monopole.  However, particle theorists like the idea of magnetic monopoles, since a lot of their theories predict them (I don't know the details of why).  If they exist, they would decay like 1/r^2, but since no one has ever seen them, most of our magnets will be dipoles, which decay like 1/r^3.  You could of course design a magnetic quadrupole or octopole.

You can, of course, design a magnetic field that behaves differently from this over a region of space.  For example, if you're near a long current carrying wire, the field goes like 1/r multiplied by the current carried in the wire (with some other constants as well), but as you get far away from it, things have to start behaving like dipoles.

#### RD

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##### Does magnetic field strength follow an inverse square law?
« Reply #2 on: 19/03/2010 12:38:20 »
IIRC the magnetic field (B) created by a current carrying wire, or a long solenoid, varies as the reciprocal of the distance (r),
i.e. B 1/r,  not 1/r2,  (not inverse square).

Quote
The magnetic field at a distance r from a very long straight wire, carrying a steady current I, has a magnitude equal to

http://teacher.pas.rochester.edu/phy122/Lecture_Notes/Chapter31/chapter31.html
« Last Edit: 19/03/2010 13:13:28 by RD »

#### JP

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##### Does magnetic field strength follow an inverse square law?
« Reply #3 on: 19/03/2010 13:19:55 »
I think this question is also about static fields.  If not, you could just take a laser pointer, and the light coming out of that (which has a magnetic field component) is very directional and isn't dying out like 1/r^2.

I don't think a long current carrying wire could exist in statics without forming it into a loop, otherwise it wouldn't be static.  You'd have to have charges draining from one end to the other and that wouldn't be static.  (You'd have a changing electric field generated by the charges at each end and the magnetic field would slowly decrease as the charges ran down).

If the wire is a loop, then you can move far enough away from it that it looks like a dipole.

Edit: By the way, I did some checking and if you can treat the system as an infinitely long cylinder (meaning that the long wire is infinitely long), then a magnetic dipole dies off like 1/r^2.  But the problem in this treatment is that I think "far away" means that you can't treat the wire as approximately infinite anymore.
« Last Edit: 19/03/2010 13:43:29 by JP »

#### chris

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##### Does magnetic field strength follow an inverse square law?
« Reply #4 on: 19/03/2010 21:33:23 »
Thanks for the careful analysis of this.

I started looking into it because when we discussed wireless chargers on the radio programme I suggested that this technology is only useful over short distances; the reason I put forward was that the base coil (the driver) produces an alternating field that dies away according to 1/r^2 and that therefore the slave coil - resident in the device and which picks up the power from the changing field - needs to be sufficiently close to feel the changing field and hence generate a charging current.

Should I have said, therefore, that the slave coil sees a magnetic field that decays as 1/r^3 ?

Chris

#### LeeE

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##### Does magnetic field strength follow an inverse square law?
« Reply #5 on: 19/03/2010 21:56:18 »
Just to briefly mention that I recently heard about a low-speed electrically powered personal transport system where the power for the electric motors was induced, from cables buried in the road, the vehicle itself carrying no batteries or power storage.

I think the same issues applied though, and the system isn't very efficient.

#### RD

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##### Does magnetic field strength follow an inverse square law?
« Reply #6 on: 19/03/2010 23:33:20 »
I started looking into it because when we discussed wireless chargers on the radio programme I suggested that this technology is only useful over short distances; the reason I put forward was that the base coil (the driver) produces an alternating field that dies away according to 1/r^2 and that therefore the slave coil - resident in the device and which picks up the power from the changing field - needs to be sufficiently close to feel the changing field and hence generate a charging current.

Allegedly "magic" resonance is involved in this wireless power transfer technology ...

Quote
At first glance, such a power transfer is reminiscent of relatively commonplace magnetic induction, such as is used in power transformers, which contain coils that transmit power to each other over very short distances. An electric current running in a sending coil induces another current in a receiving coil. The two coils are very close, but they do not touch. However, this behavior changes dramatically when the distance between the coils is increased. As Karalis, a graduate student in electrical engineering and computer science, points out, "Here is where the magic of the resonant coupling comes about. The usual non-resonant magnetic induction would be almost 1 million times less efficient in this particular system."
http://web.mit.edu/newsoffice/2007/wireless-0607.html
« Last Edit: 19/03/2010 23:44:53 by RD »

#### JP

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##### Does magnetic field strength follow an inverse square law?
« Reply #7 on: 20/03/2010 00:08:00 »
Ah.  In that experiment, they're pumping energy into the system  so that it radiates an electromagnetic field rather than having a static field around itself.  For radiating fields, both the electric and magnetic fields should die off like 1/r^2, so you were right.  The 1/r^3 rule for magnetic fields arises in the case of being really far away from static (unchanging) magnetic fields.

#### chris

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##### Does magnetic field strength follow an inverse square law?
« Reply #8 on: 20/03/2010 09:15:11 »
Thank you JP - Phew - I feel both relieved and vindicated!

Is the radiating field decaying by 1/r^2 because it's effectively light?

Also, what would be the correct way to calculate the energy transferred from the primary to the secondary coil in the wireless charger at distance x?

Cheers,

Chris

#### JP

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##### Does magnetic field strength follow an inverse square law?
« Reply #9 on: 20/03/2010 09:41:29 »
Yes--the decay of 1/r^2 comes from it basically being light-like electromagnetic radiation.  It sounds like they're using a much lower frequency, however.  The calculation of energy transfer is usually done by looking at the time-varying magnetic fields within the receiving coil, which induce a current (this is called electromagnetic induction).  A simple way to think about this is that the current induced in the wire is such that the wire generates a magnetic field that opposes the change of the external field.  (In other words, the total field inside the wire loop "tries" to stay constant, and this creates currents in the wire loop.)

However, what RD was pointing to above means there's something a bit different going on in this setup.  They say they can beat the usual rate of energy transfer due to induction by using resonances between the transmitter and receiver.  It's hard to tell more without having more technical information on their setup.  They might beat the 1/r^2 law, however, since that assumes power is being transmitted equally into all directions and the resonance might end up making it a more directional energy transfer.

#### Geezer

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##### Does magnetic field strength follow an inverse square law?
« Reply #10 on: 21/03/2010 01:58:43 »
I would think the energy receiver has to behave like a "tuned circuit". IOW, it's an antenna that is optimized for a particular EM frequency.

Of course, that may be baloney!

BTW, I think some researchers have created antennae that can receive RF in the visible wavelengths. The structures are rather small.

#### RD

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##### Does magnetic field strength follow an inverse square law?
« Reply #11 on: 21/03/2010 03:10:17 »
Their wireless power transfer looks distinctly directional to me ...

Quote
Wireless power transfer over two-meter distance, from the coil on the left to the coil on the right, where it powers a 60W light bulb. Members of the team that performed the experiment are obstructing the direct line of sight between the coils.
http://web.mit.edu/newsoffice/2007/wireless-0607.html

BTW If a Hollywood movie is made of this 'scientific breakthrough' Kelsey Grammer is ideal to play Professor Joannopoulos  ...

« Last Edit: 21/03/2010 04:58:52 by RD »

#### Geezer

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##### Does magnetic field strength follow an inverse square law?
« Reply #12 on: 21/03/2010 04:34:09 »
BTW If a Hollywood movie is made of this 'scientific breakthrough' Kelsey Grammer is ideal to play Professor Joannopoulos  ...

Looks like a "dead ringer" for his brother. Wait a minute - isn't that the punchline for an old joke?

(There are actually two punchlines.)
« Last Edit: 21/03/2010 04:39:18 by Geezer »

#### RD

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##### Does magnetic field strength follow an inverse square law?
« Reply #13 on: 21/03/2010 04:57:20 »
It's hard to tell more without having more technical information on their setup.

http://www.mit.edu/~soljacic/wireless-power_AoP.pdf

BTW
Quote
This work was funded by the Army Research Office (Institute for Soldier Nanotechnologies),
National Science Foundation (Center for Materials Science and Engineering), and the Department of Energy.
http://web.mit.edu/newsoffice/2007/wireless-0607.html
« Last Edit: 21/03/2010 14:58:50 by RD »

#### Geezer

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##### Does magnetic field strength follow an inverse square law?
« Reply #14 on: 21/03/2010 05:09:35 »
They could have saved a few bucks at the receiver by using much thinner wire.

I'm not sure Marconi would have been all that impressed.

#### Bored chemist

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##### Does magnetic field strength follow an inverse square law?
« Reply #15 on: 21/03/2010 09:58:07 »
I wonder what these folks
http://www.hpa.org.uk/webw/HPAweb&Page&HPAwebContentAreaLanding/Page/1153822623782
(or their local equivalent) would think of that demonstration.

#### RD

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##### Does magnetic field strength follow an inverse square law?
« Reply #16 on: 21/03/2010 15:24:44 »
I wonder what these folks
http://www.hpa.org.uk/webw/HPAweb&Page&HPAwebContentAreaLanding/Page/1153822623782
(or their local equivalent) would think of that demonstration.

Depends on whether they guy in the red shirt had Bell's palsy before he went between the coils.
« Last Edit: 21/03/2010 15:27:19 by RD »

#### JP

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##### Does magnetic field strength follow an inverse square law?
« Reply #17 on: 22/03/2010 06:35:30 »
Ok, I browsed the article quickly.  They're not using what we traditionally call radiation, which consists of waves carrying energy away from the source in the form of waves.  For traditional radiation, the power being transmitted in a given direction dies off like 1/r^2.  Instead , they're using what are called evanescent waves, which decay exponentially and are usually ignored since they die off in such a short range that they don't matter for most applications.  In the systems here, it seems that they've made their setups such that the evanescent field dies off relatively slowly in a specific direction.  They then take advantage of a really efficient way to transfer energy via the evanescent fields (the resonant coupling they describe in the paper, which I admit I don't fully understand).

So in this case, it seems you're limited to relatively short ranges due to the exponential decay of the evanescent field rather than the radiation falling off like 1/r^2, though they say they've engineered the systems and chosen microwave wavelengths so that ranges on the order of meters are obtainable.

#### Farsight

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##### Does magnetic field strength follow an inverse square law?
« Reply #18 on: 22/03/2010 12:20:02 »
Very interesting, evanescent waves.

Chris, you can get an intuitive answer to your OP by considering a thin sheet of rubber. Stick a pencil point under it and push upwards, and the field diminishes in all directions - north south east and west and everything in between. If instead of one point you push a long line of contiguous points up into the sheet, the field can only diminish in the east-west direction, so it extends further. This is akin to the current in the wire. Next if you push a circle of points up into the sheet, the field inside the circle is uniform, and doesn't diminish with distance. This emulates a solenoid. This analogy isn't perfect, since the field outside a solenoid is scant, but it gives the gist in quite a nice way. So:

Q: does the strength of a magnetic field, measured at increasing distance from the field source, decay according to 1/r˛?

A: for a single source such as an electron, yes. The magnetic field is how we describe one aspect of the electromagnetic field when in relative motion with respect to the source. But the disposition of multiple sources can result in a magnetic field which doesn't follow this rule.
« Last Edit: 22/03/2010 12:22:41 by Farsight »

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##### Does magnetic field strength follow an inverse square law?
« Reply #18 on: 22/03/2010 12:20:02 »