This might do better in the electrical part of the forum but   here are some initial thoughts.

The standard you quote is for LV installation  i.e.  50V to 1kV so not really applicable-  24V would be  ELV, and a bathroom cabinet sounds like an appliance, not an installation, for the same reason not really in the BS7671 domain either.

It probably can be done in some way, but there are some considerations.

I'd hope the current to be limited in a safe way, so there is no fire or damage risk if the user connects the hinges with something metallic, as this is a credible user action.

How is the 24V derived - it needs to be SELV, so both 'floating' and isolated relative to the low voltage mains.

How many watts are being supplied, and are the hinges the sort with a ball race ?  if so arcing and burning damage to the bearings may be an issue at high currents, though I'd expect an plain steel hinge to be OK to a few amps, brass rather more. If you wanted a kW I'd be very wary indeed.


Mike.

PS
HSG85  as an example considers exposed conductors to be a hazard  at 50V and above, or if the available current is high, and then refers out to PD 6519 for the shock levels.

Live parts that are uninsulated and exposed so that they can be touched either directly or indirectly by a conducting object are hazardous if the voltage exceeds 50 V ac or 120 V dc in dry conditions ; see BSI publication PD 6519 (3) ; and/or if the fault energy level is high; 

The implication is that any exposed voltages a lot less than this, with current limit, are not really hazardous, in the eyes of the HSE. The key may be the dry conditions, in a bathroom folk may be wet and naked, less likely in a factory setting.

This might do better in the electrical part of the forum but   here are some initial thoughts.

The standard you quote is for LV installation  i.e.  50V to 1kV so not really applicable-  24V would be  ELV, and a bathroom cabinet sounds like an appliance, not an installation, for the same reason not really in the BS7671 domain either.

It probably can be done in some way, but there are some considerations.

I'd hope the current to be limited in a safe way, so there is no fire or damage risk if the user connects the hinges with something metallic, as this is a credible user action.

How is the 24V derived - it needs to be SELV, so both 'floating' and isolated relative to the low voltage mains.

How many watts are being supplied, and are the hinges the sort with a ball race ?  if so arcing and burning damage to the bearings may be an issue at high currents, though I'd expect an plain steel hinge to be OK to a few amps, brass rather more. If you wanted a kW I'd be very wary indeed.


Mike.

PS
HSG85  as an example considers exposed conductors to be a hazard  at 50V and above, or if the available current is high, and then refers out to PD 6519 for the shock levels.

Live parts that are uninsulated and exposed so that they can be touched either directly or indirectly by a conducting object are hazardous if the voltage exceeds 50 V ac or 120 V dc in dry conditions ; see BSI publication PD 6519 (3) ; and/or if the fault energy level is high; 

The implication is that any exposed voltages a lot less than this, with current limit, are not really hazardous, in the eyes of the HSE. The key may be the dry conditions, in a bathroom folk may be wet and naked, less likely in a factory setting.