Flirting with Ohm's Law

I had a very nice halogen floor lamp that we moved here from the US. I had a vague plan to covert it for European voltage. After three years, I finally managed to do something about it. It has a dimmer switch on the floor, and I considered the possibility of just rewiring the cord going into the dimmer. It seems most light bulbs are half the watts as in the US--- and twice the voltage, of course. But I chickened out. It seems to me something is squared in Ohm's law. I didn't want to risk running twice the voltage through the dimmer, and rather purchased a new dimmer, and just reused the internal wiring. We now have the perfect lamp for the dark corner we created when we rearranged the furniture.