Area of iron core to lift a certain amount of weigh(magnet/solenoid)?
The the pull force depends only on the amp-turn product and the relative permeability of the core.

Having thicker wire only allows you to put more current in the wire.

Putting more current in the wire will create more force, but only to a point.

After the core saturates then you won't get much more force by adding current or turns.

As for coil voltage, it depends on the coil resistance.

EXAMPLE  1)  For example, suppose you had a core that was 10mm diameter and 30mm long.
Lets say you have some 30AWG magent wire that is 0.3mm diameter.

30mm / 0.3mm = 100 turns

So you could fit 100 turns of 30 AWG wire in that space.

Packed in a coil you might run 100mA through the wire.

So your total Amp-Turns is 100mA * 100 = 10 Amp-Turns

The length of the wire is (pi * 10mm)/turn * 100 turns = 3140 mm.

30 AWG wire has a resistance of 338 milli-ohms per meter.

So the total coil resistance is 3140mm * 338 milli-ohms per meter = 1.22 ohms.

So the coil voltage would be 1.22 ohms * 100mA  = 0.122V.

EXAMPLE 2)  Suppose you had the same core, but were using 40 AWG magnet wire that has a diameter of 0.08 mm.

30mm / 0.08mm = 375 turns

Packed in a coil you might run 26mA through the wire.

So your total Amp-Turns is 26mA * 375 = 9.75 Amp-Turns.

The length of the wire is (pi * 10mm)/turn * 375 turns = 11,780 mm.

30 AWG wire has a resistance of 3.44 ohms per meter.

So the total coil resistance is 3140mm * 3.44 ohms per meter = 40.5 ohms.

So the coil voltage would be 40.5 ohms * 26mA  = 1.053V.

Note that in both examples size of the core was the same and the Amp-Turns about the same (10.0 vs 9.75), so you would get about the same pulling force. But the coil voltage was different bcause one coil resistance was different.

As for the exact pulling force it is highly dependent on geometry and your core material.  You may just need to experiment on that one.



Note that in this case the 30AWG wire used 100mA * 100mA * 1.22 ohms = 12.2mW of power.

The 40 AWG wire used 26mA * 26mA * 40.5 ohms = 27mW.

So in this case the 40AWG wire was less efficnet.  The reason is that there was less total volume of copper (since the layer of 40AWG wire was thinner).
If we used four layers of 40AWG wire the height of the four layers would total to 0.32mm (30AWG was 0.3mm).

Making an approximation (since the wire diameter is small compared to core diameter)

For four layers of 40 AWG wire we have 375 Amp-Turns * 4 = 1500 Amp-Turns.

The coil resistance would be 40.5 ohms * 4  = 162 ohms.

To get 10 Amp-Turns we would need 10 Amp-Turns / 1500 Turns = 6.7mA.

The coil voltage would then be 6.7mA * 162 ohms = 1.085V

So clearly for a given wire size and Amp-Turns the coil voltage did not change when we increased the number of turns.

6.7mA * 6.7mA * 162 = 7.3mW.

So now the 40AWG wire is more efficient than the 30 AWG wire because we increased the volume of copper in the 40 AWG case.