I don't think the Wii scale uses the electricity. It seems to do a BMI calculation based on your weight and the other info. you entered (height, age, etc.). That means it's probably not the most accurate measure, but you also don't have to worry about electricity.
I am going to be in Chicago for the weekend, which means I will miss two days on it. I'm really going to miss my workouts (can't believe I'm saying that). I dread how it's probably going to yell at me too...when I gained a couple pounds in a day, it made me enter the reason. Oh well, accountability is good, even if it's just to a video game.
ETA: Here is how it calculates BMI. It's just through info., no electricity:
After reminding you to stand up straight, the game will ask if you're wearing any heavy clothing. Then it will determine where your weight is distributed. More than likely your beef won't be perfectly centered so you will be shown where your center of gravity is by a red dot on the onscreen balance board. You'll need to shift your weight so that the dot is placed right in the middle of the board. This will likely feel awkward, but you don't need to keep that stance for long. The game apparently just wants to demonstrate what terrible posture you have.
After this, Nintendo has enough information to calculate your BMI. We should note that Wii Fit does not determine your weight. It determines your body's bulk in relation to your height and lets you know if you've got too much mass going on. A vertical meter is separated into blue (under weight), yellow (just right), and red (over weight) segments, and an arrow will indicate where you fit in. Then, perhaps comically, the size of your Mii will be adjusted to match your BMI.