It is simply an ohmic heating issue. If the wire is in a transformer, the peak value is less still; if the wire is in the open air as an overhead cable, it might be more. I doubt if there is any good formula to calculate it as it would depend too much on what surrounds the wire and its temperature and thermal conductivity. The value you quote is a good average.

It is simply an ohmic heating issue. If the wire is in a transformer, the peak value is less still; if the wire is in the open air as an overhead cable, it might be more. I doubt if there is any good formula to calculate it as it would depend too much on what surrounds the wire and its temperature and thermal conductivity. The value you quote is a good average.

The current density is related to the temperature of the conductor . the current density is not restricted to 500 A/sq.cm - it is related to the temperature which in turn depends upon the  cooling medium  .

The temperature of the copper conductor is to be restricted to 225 deg C  beyond which the material is likely loose it's mechanical property . 

In oil filled equipment , the current density can go beyond 5A / sq. cm if the temperature is below 225 deg C .

In case of machines and transformer design assumes 5.5 A/mm2 as astarting design value.

The transformer has to see many types of forces and cooling is also a concern .

 So, the value of current density in copper conductor  in coil is kept around 2.5 - 3.5 Amp/sq.mm .

However , the value of 5.5 A/sq.mm is not an un-natiural value for other current carrying copper parts  .

Any design current limit for Cu is driven by the rise of temperature due to resistive losses.

Here in Germany, 16 A are the limit for 1.5 mm2 3-wire installations, equating in beyond 1000 A/cm2. But for denser wire 'packing', a reduction is necessary.

Some considerations are given in ISO standards covering current carrying capabilities. (Sorry - cannot remember any ISO 'number' on this topic. But you should be able to find them with these terms.)

Regards

To avoid heating, and insulation detoriation , yet short time over loading for equipments are done, refer my papers......

In addition to the Resistive loading , the conductors also get a loading owing to the interaction with the electro-magnetic field and it is difficult to determine the exact loading . Hence , some empirical guide lines are provided based upon the particular use . However ,  the prime concern is the behavior of  material property i.e mechanical strength .

Another concern is the presence of voids , in case of joints in the conductor - the voids expand under the effect of temperature . So in total , it is the temperature that govern the current density .

Nice...!

This back to the electrical properties, electrical conductivity and spacing between molecules. 

There are many considerations , which otherwise limit the current density . For example - the machines which are susceptible to forces , need to have higher conductor dimensions which automatically restricts the current density . The machines that are subject to high frequency ( harmonic ) have to have smaller cross section that automatically tends to have higher current density . The underground cables have lower current density than over head cables . The oil filled machines have higher current density as compared to air filled machines .The AC machine have a tendency to have lower current density as compared to DC machine . So , current density is a derived parameter based upon the actual functional requirement . Even though there is no direct mathematical formula to restrict it , it gets limited by of the requirements .

If the wire is in a transformer, the peak value is less still; if the wire is in the open air as an overhead cable, it might be more. I doubt if there is any good formula to calculate it as it would depend too much on what surrounds the wire and its temperature and thermal conductivity. The value you quote is a good average.

Current density 5A/mm2 or 500A/cm2 is taken based on the experience of reliable operations without much/no heating.

By increasing the current density further may increase the copper losses and thereby heating, which damages the insulation properties.

However if active cooling is provided current density can be increased further without failure. Also if the same conductor is used in encloses environment or vacuum then the current density will be less.

I heard different values (in the range of ~1-10 A/mm^2) for the maximum current density of the Copper wires depends on the applications.

From the above, it's obvious there's no simple answer here. That's why I prototype my parts with thermocouples inserted within the windings. I have seldom found the Current Density above this that works, most are lower by far. One exception was a Helical Wound, low Inductance we used in a PSFB converter.