You will not pay interest on $1. You will pay interest on your Average Daily Balance.
Say you charge $20K and then pay off $19999 on day 15. Your ADB is (($20K * 15) + ($1 * 15))/30 ~= $10K. Multiply that by your monthly interest rate and that's what you'll get charged in interest. You'll see this hidden in there if you read the fine print on how finance charges are calculated.