If there is a given change in current dI/dt in a wire and I want to
know what emf will be induced in another wire at a distance x , how
can I estimate it if there is only air between the wires?

At the moment the nearest thing I can find is the equation e = -M
dI/dt, which I understand relates induced EMF e on one coil to the
change in current on another coil in a transformer, where M is the
'mutual inductance'. This isn't much good to me, as I have no way of
working out the mutual inductance the sort of system I described
above. I also have an idea that there's an inverse square relationship
between distance and induced emf involved, as well.

I would be particularly interested in any way I can work this out
without using vector calculus, as I'm not familiar with it yet. If
there's no alternative, could someone please explain what the notation
means, as well?

Formula for mutual inductance is simple:

it is given here 
http://farside.ph.utexas.edu/teaching/302l/lectures/node92.html
or here
http://homework.phys.utk.edu/courses/summer2002/phys222/exam2.html

Search Term is: Lenz Law

tricky part: You must pay attention to the  geometry - 
which means e.g. this

Two coils  (of given crossection, distance, angle ..) are different from
two parallel (infinite) wires 
(which do not interact by square, but by linear inverse).

Example of that, effect of geometry, is discussed e.g. here:
http://www.newton.dep.anl.gov/askasci/phy00/phy00152.htm

This page shows mutual inductance of two coils, which has a common path for
magnetic flux:
http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/indmut.html

This page defines mutual inductance in general:
http://farside.ph.utexas.edu/teaching/302l/lectures/node84.html

Here is an (aproximate) formula for two loops in air:
http://www.sigcon.com/lib/htm/mloop.htm

Interaction between two wires is described here:
http://iml.umkc.edu/physics/wrobel/PHY250/lecture6.pdf

The general formula -for any shapes - is derived by calculus
by summing the contributions from pieces of the two objects.

Please, do review above links, and feel free to ask for clarification
if needed.

H

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Request for Answer Clarification by rexregum-ga on 04 Oct 2004 09:10 PDT

How complicated is it to derive the approximation on the page listed below?

http://www.sigcon.com/lib/htm/mloop.htm

On reflection, it's exactly what I needed, and I'm very pleased that
you found it. It made me realise that I was actually asking slightly
the wrong question.

If it involves contour/vector integration, I'll stop here, but
otherwise I would like to have an idea of how it can be derived.

---

Request for Answer Clarification by rexregum-ga on 04 Oct 2004 09:43 PDT

One slightly more serious point; the input units for that
approximation appear to be inches. I believe that if I convert all
units into inches [and inches^2] first, it should give me the correct
SI units of inductance to go with the equation e = -M dI/dt [Henrys, I
think]. Please confirm that this is so.

---

Clarification of Answer by hedgie-ga on 04 Oct 2004 20:42 PDT

1) aproximation
 Expression is derived from Faraday Law (ultimately from the Maxwell Equations)
 -in principle by summing over the elements of the loop 
that is by integration (but not by contour integration)

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html

 Another way to arrive to it is to use multipole expansion
 (which of course includes vectors)
http://scienceworld.wolfram.com/physics/MultipoleExpansion.html

 The conceptual approach is this:
The mutual inductance (which also enters
expression for mutual force) depends on exact shape of the
loop. Each tiny piece of coil A is affected by the field
(created by B) and that field depends on exact shape of B.

However, at large distance from B, field looks like a field of
a dipole -- details of shape can be neglected 
that is shown here

 http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html


So you can imagine two rod magnets (two dipoles) which atract each other
- and at large distance their shape is not important - only their moments
matters.
2)units 
   The 'real' physics equations work in any consistent system of units.
(that is a search term) - meaning equations do not contain conversion factors.
http://www.csee.umbc.edu/help/theory/units.shtml

I strongly suggest you do calculation in SI and if necessary, 
convert the result into whatever. 
In this case  (SI units) you get result in Henrys
http://www.roymech.co.uk/Useful_Tables/Units/DerSIUnits.htm

3) I feel you had a bad experience with vectors and countour integrals
and I want to assure you that it is not vectors fault nor your fault.
Most likely it was  just a bad and unsuitable textbook. 

Take your time and find one which is right for you
ftp://joshua.smcvt.edu/pub/hefferon/book/book.pdf
http://archives.math.utk.edu/topics/linearAlgebra.html
...
  and you will find out that vectors/linear algebra/tensor calculus
 is beautiful field which make perfect sense.

My favorite intro is: 
http://www.amazon.com/exec/obidos/tg/detail/-/0486655822/002-1451747-8990406?v=glance

but - everyone has to find the one which is right for him (or her)
 (I mean book )


Good luck

Hedgie

Thank you for the advice on contour and vector integrals; the reason I
didn't want to deal with them is that I'm only 17 and I'm currently
quite busy learning lots of other maths [exam in < 1 month]. In other
words, I've haven't used them before and I don't really have time to
learn just now. But thank you nonetheless.

I've checked the units, and I get realistic answers when I convert
from SI to inches, so I'm happy with that, too.

Goodbye, and thank you for all your help.

Can?t remember off the top of my head but can explain the inverse
square law if that helps a bit.

If you have 3D point radiator, the field will drop off as an inverse
square with distance purely by geometry. Double the radius of a sphere
and the area will be four times the original. Same applies to any 3D
shape, such as a conic section or even a square projection. So the
formula, where a = area and d = distance is :-

a2 = ((d1/d2)squared) x a1
Rearrange as required of course.

Interesting point which may occur to you -- how then do dish aerials
seem to bypass the square law? They don?t, but in essence the geometry
puts the point radiator point way behind it. That?s the point from
which to calculate inverse square from.

Pictures for ?inverse square law? :-

http://hyperphysics.phy-astr.gsu.edu/hbase/forces/isq.html
http://csep10.phys.utk.edu/astr162/lect/light/intensity.html

If no one answers and I find the time........

Best

Thanks for the advice; it'll certainly help. Which other researchers
are likely to be able to answer this question?