Amps * Volts = Watts
Amps * 12 = 600
600/12 = Amps = 50 amps
You would need a reserve capacity, so I'd go somewhere between 60 or 100 Amp rated transformer.
Transformers are rated in volt-amps which is usually calculated the same as watts. But the term "watts" technically does not apply to transformers. So you need a 600 volt-amp transformer or, as Redbeard has suggested, you need an 800 or 1000 volt-amp transformer.
That's a lot of amps for a 12 volt system so I recommend you double check your requirements. You will need a #2 gauge wire if your requirements are correct.
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A 60VA transformer can run only one 50W bulb.
Answer for countries in Europe and other world areas running a 50 Hz power supply service. This depends on the the light or globe, if you can replace just the globe its very simple. change over the globe and buy yourself a mains(230v or 115v) to 12v isolation transformer, and attach that to the mains cable and run a cable from the 12v to the globe and away you go. The description of how to change the mains to 12v is the same if you have to replace the whole light. Then you just have to replace the globe or the whole light assembly.
No; drawing more than the rated amperage from a transformer will cause it to overheat.
The power lost by hysteresis depends on the peak flux density in the core. If the transformer is getting hot even when on no load, it should be run at a lower voltage.
As well as providing a cooling medium, transformer oil is also used to insulate a transformer. As its dielectric strength (how well it insulates) is much higher than air, it means that components within a transformer that are at different potentials can be placed much closer together if they are immersed in oil. So, operating an oil-filled transformer without oil will likely cause insulation breakdown within the transformer.