The volt-drop of the length of Ethernet cable may be enough to allow the normal modem to work at 12V, if 24V was feed from the POE switch.

I will do some tests on my POE access point when it comes that I have just brought.

Volt drop doesn't work like that. 100 metres of CAT5 / 5e has a loop resistance of ~20 Ohms per 100 metres (of course, the maximum permitted run between ports is 91 metres in the CAT 5 / 5e specification). Given that the modem doesn't draw a constant level of power whilst in operation, it would be impossible to regulate the upper voltage limit with a fixed length of cable (for an explanation of this, see Ohm's Law). In addition, the cable would need to be longer than the maximum data transmission limit given the port power output of 15.40W as specified within IEEE 802.3af - and make the wiring quite warm too.
See http://www.belden.com/docs/upload/PoE_Basics_WP.pdf

Edited by deleted (Wed 18-May-16 18:50:54)