Take this with a grain of salt since I do not have white background. All I have is gray. If you already have white background, you probably dont really need to add light to the background. If you do, you don't need much. You just have to make sure you are not casting shadow on the background. More than likely you will have pure white anyway. I think the light to the background bounced back and hit your camera. That is why it looks a little low in contrast.
Now if you are shooting on a gray background like me, that is when you really need the background light to make the background look white. So if I want white, add light. If I want black, I block the light hitting the background. If I want gray, add light a little. I didnt do it to the extreme.. but you get the idea. These are both shot with same gray background. I could have made the background more powerful on the 2nd one and get pure white.