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For a pair of long, straight wires carrying current in opposite directions, if the distance between you and the wires is significantly greater than the distance between the pair of wires themselves (which is almost certainly the case if the wires are barely a millimetre apart and you're not touching them), then the magnetic field strength in *amperes per metre* (the H-field in Maxwell's equations) is given by:

Twice the current flowing in the wires (in amps), multiplied by the wires' separation distance (in metres, probably about 0.001), divided by the square of the distance between you and the wires (in metres).

You can get this from a paper in the IEEE Transactions on Space Electronics and Telemetry, Volume 10 number 4 (December 1964), pages 154-158, where we read how NASA's Alberta Alksne worked out how much magnetic field would be coming from the wires in a spacecraft. Looking at his Equation 7, the two components of the field are calculated by:

- the current flowing in the wires
- multiplied by the wires' separation distance
- divided by the square of the distance between you and the wires
- multiplied by:
- for the X direction, -4 *
*cos t***sin t*(because, for some angle*t*,*x=r cos t*and*y=r sin t*, so*xy/r^2*is just*cos t***sin t*, and Alksne's wire-separation distance is*twice*the value of*a*, see his Fig. 1, so we halve the -8 to compensate), - and, for the Y direction, +2 - 4*(
*sin t*)^2 (because r^2=x^2+y^2 by Pythagoras, so x^2-y^2 = r^2 - 2y^2, so we have +2 * (*r*^2-2*(*r sin t*)^2)/*r*^2 and the*r*cancels out upon simplification), - so the multiplier for the overall magnitude is sqrt((4 *
*cos t***sin t*)^2 + (2 - 4*(*sin t*)^2)^2) - and it can be shown (numerically if you like) that this is always equal to 2 no matter what the value of
*t*.

- for the X direction, -4 *

Alksne's Fig.2 shows that the worst possible case is when the length of the twist is about 10 times the distance between you and the wire. In this case you might need to add nearly 10% extra to the field strength on the X axis, although it drops in other directions and with other twist combinations. So twists *usually* work in your favour, but if we're being really pessimistic, we'd better say you might be 'unlucky' enough for the wires to be twisted in such a way that 10% is added, so we have:

2.1 * amps * about 0.001 / distance^2although that extra 10% is probably within the error bars of the "about 0.001 separation" anyway, unless you pull apart your wiring to measure it exactly (please don't!)

teslas = 2.64e-9 * amps / distance^2and, since a tesla is about 10,000 gauss,

gauss = 2.64e-5 * amps / distance^2or, since discussions of EMF health risks usually refer to

milligauss = 0.0264 * amps / distance^2

As we seem to be just fine in the Earth's "DC" magnetic field of 250 to 650 milligauss, which would take a massive 10,000-25,000 amps to match using DC current in a 1mm-separated pair of contraflow cables a metre away (don't try that at home---it'll burn the wires and possibly your house, which is *definitely* a bigger health risk than the magnetic field), I assume the only thing we *might* have to worry about (with pairs of cables in contraflow) is the "AC" part of the field.

Phone chargers typically take more current during the phone's "fast charge" phase and less in its "trickle charge" phase. The new "Quick Charge" chargers are supposed to get the fast phase done in *minutes*, which makes them less likely to be relevant in "charging while you sleep" calculations, so let's look at older chargers' fast phases. If the charge is taking place over a USB cable then the output voltage will be 5V (USB standard), and the output current is typically a maximum of 1A (but iPad chargers can do 2A), which means 5 (or 10) watts; since the charger is inefficient, you probably have to double this to 10 (or 20) watts on the mains input side, which at 240V gives 0.04 (or 0.08) amps; it'll be double that in America where the mains is only 120V (watts = volts x amps). Therefore, for the *input* to the phone charger, we're looking at:

- 0.001 (or 0.002) milligauss for a 1mm-separated 240V wire at 1 metre (Americans have to double this for 120V)
- up to 0.008 milligauss if that wire is only 50cm away
- up to 0.2 milligauss if that wire is only 10cm away
- double again if it's separated by 2mm instead of 1mm
- double again if it's a big one separated by 4mm

But what about the current coming *out* of the phone charger, which has a lower voltage but higher current? It's *supposed* to be DC, but it *can* also have ripples or (if the charger is a cheap half-wave rectifier) pulses. I don't know if a *pulsed* DC field will be as bad as an AC field, but let's assume it is and check:

- for a 1-amp charger, 0.026 milligauss at 1m (double for 2-amp, but hopefully a 2-amp charger won't be so pulsed)
- increasing to 0.1 milligauss at 50cm, and 2.6 milligauss at 10cm

I'm not absolutely sure how that conference arrived at their "1 milligauss" figure, but I think it's related to a 2008 study of Chinese children (Yang et al, Leukemia & Lymphoma 49(12):2344-2350) which looked at 123 patients and 135 healthy children and said the patients tended to be getting 1.4 milligauss from power lines near their houses, although nobody seemed to be made worse by communications transmitters (which run on a different frequency) or indoor appliances. (A previous study they cited by Maslanyj et al put the threshold at 4 milligauss and said indoor appliances account for *most* of it, so we can see where the "it matters more when you're asleep" idea might have come from. Tissue repair happens more during sleep, and the bedroom appliances mentioned in the Chinese study---microwave ovens etc---are not typically in use during sleep.) Unfortunately Yang et al didn't have the resources to quantify what percentage of *everyone living near power lines* got cancer, as opposed to what percentage of everyone *not* living near power lines did so, so the study still doesn't tell us *how big* a risk factor the power lines are, but it does seem to show *some* risk (although the original data doesn't seem to be available so we can't check the exact details of their statistical tests).

I've separately seen the figure "0.2 milligauss" mentioned as a slight concern, but I haven't seen anything concrete as to how this was arrived at. (The human brain itself produces a changing magnetic field, but only in the nanogauss range; that doesn't mean we can cope *only* with nanogauss though.) If 0.2 milligauss is correct, then you can be concerned about sleeping 10cm away from a mains cable feeding a phone charger, but greater distances should still be OK.

So generally, a cable (whether mains or low-voltage) that's involved only in charging a phone should be fine to sleep near to if it's more than 10cm away.

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