That would depend on what sort of statistical analysis you intend to do with your data. Sample size and power of a test are directly related: more is usually better, especially if differences are subtle. If you're really unsure, talk with a researcher or stats person who can help you determine what you want your survey to find out, how to analyze your data, and from that pull out what sorts of results you think you'll see. This should give you a better idea if it should be 10, 100, or 1000. (But I can tell you already it's not going to be 10, and good luck getting a decent response rate: doctors are notoriously bad survey responders.)
If you're doing sponsored research in conjunction with your school, talk with your advisor. You'll need the affiliation for publication anyways, and likely a run through a human subjects committee.