Electronic – Optimizing magnetic field force of a coil

Update from OP.
Current: Needs to be no more than 30mA.
Distance: A 4mm air gap.
Minimum data rate: Around 256Bits/second. Size: Needs to be as small as possible.(Must fit into 5x10x5mm spot)
Cost: Looking to keep it under $1.50

Is that 30 mA receive or 30 mA transmit.
Unidirectional?
Cost of $1.50 covers what? TX & RX, just one (which?),Hall cell in that price.
How many? 1 10? 100? 100,000?
MUCH more information allows us to provide a single instant answer without playing death of 1000 cuts / iterations.

The Hall cell chosen is completely unsuitable for this task.
This is because it is a sampling type which sleeps for most of the time and wakes to take a reading occasionally.
Th data sheets hows that it has a 0.1% on time and 99.9% off time.
Cycle time is 45 to 90 ms and on time is 45 to 90 uS.
So you can only signal at most at 1 bit per on time if you are careful or at about 10 bps max and probably less.

There are many Hall cells available which are not the sampling type and low enough current.

[This is Digikeys cheapest at about 58c/1.]http://www.semicon.toshiba.co.jp/docs/datasheet/en/Sensor/TCS20DPR_en_datasheet_110207.pdf)
This has 4.4 mT sensitivity worst case.
Mutiply T by 10,000 to get Gaus.
4.4 mT x 10,000 = 44 Gauss = about te same as before.

Doable at range and size specified. Implementation details depend on all answers not yet known.

More when more known ...

This question is eminently answerable but rather than giving you a single "this will work" answer, having more information will lead to a much better answer. What range do you want to work over from the face of the Hall cell to the face of the inductor?
Is there anything in the way obstructing, spinning, cutting ...?
Is it in seawater, embedded in a block of steel or a lava field, ...?
What maximum data rate do you require?

Be as specific as possible re constraints on cost, size, and anything else you can hink of. DO NOT have us say xxx meets your needs and then say "Oh, but it must be British Racing Green and work at 2000 feet underwater" or whatever :-)

Don't let the following worry you. The answer is a piece of ferrite and some wire - but this is "what lies underneath":

IF there is a need to wind a coil and activate the sensor at a distance, then it may come down to formulae like this:

Relating to an inductor like this:

(http://www.netdenizen.com/emagnet/solenoids/thinsolenoid.htm)
From here

Or it's big brother which has finite thickness, from here

BUT probably not.

Adding a core increases the field by the permeability of the core - but, we'll come to that.

FWIW those formulae are about the nicest I've seen for a common problem that usually get's a horrendously complex answer. This is mainly geometry. Most analyses are for the field INSIDE the solenoid and few deal with it beyond the ends.

A simple model to help you visualize: The magnetic flux forms loops, you cannot have a magnetic field line terminate on nothing in free space. That means that all those flux lines inside the solenoid heading (for example) to the right. The flux line must continue outside the solenoid and loop around to join the other end of the flux loop. In doing so it heads to the left. No external flux lines -> no flux.

So what could be the issue?

1) In an infinite solenoid, there is no external magnetic field. One example of this is a Tokamak or toroidal solenoid. Like a big doughnut. The flux lines form loops within the doughnut and have no need to go outside. - of course a mathematically ideal infinite solenoid could have been what they were discussing, in which case the field lines loop around infinity. Hard to test though.

2) The flux density outside of the linear finite solenoid is much less than inside. So while there may be X flux across the throat of the solenoid that same amount of flux (X) is spread out over a much much larger cross sectional area. The density is much less. [here the area is the cross sectional area across the solenoid.]

3) Perhaps they are talking about a shielded solenoid? This would involve having a high permittivity material external to the coils to trap the flux lines.