Its not really a shower cable size problem - although it could be if the cable gets warm when the shower is being used......Most houses are fed with a 60 amp supply of electricity. If your shower uses say 36 amps or more, that doesn't leave a lot left over...especially if there are other appliances in use in the house at the same time. Another way of preventing this power drop is to have your lights fed by two circuits instead of one...although it doesn't cure the problem it certainly helps it.
John
July 2009