Question: So here is my problem, I'm an intern at a fairly large offices. My supervisor and manager are out of the office for at least another 2 weeks and I've finished the work they have set me. Usually I wouldn't be hugely busy but now I am literally sitting at my desk more or less twiddling my thumbs, or trying new programming languages. My manager asked my team members before he left to give me work if they had any to give, but I have yet to receive anything. 

So my question is how could I phrase asking for work when going around my department that wouldn't come off as desperate? Or what kind of way could I write a group email?

Answer: 
Asking for work is never seen as desperate. Quite the opposite, it is seen as proactive. 

You first go to your colleagues, and if they have nothing for you, then you just walk to the nearest manager. Most have plenty of work that would be useful but not useful enough to justify spending manpower on, so if you offer to work for them for free (it’s free to them) they will take you up on that offer.
Finishing your work earlier than expected (*while still performing well*) is seen as something nice. Your supervisor will be happy with this!

What I would do, is **send a quick email to your supervisor** (and possibly your manager too, depending on who keeps an eye on whatever you do.) telling him you finished early, solved the problem and will be asking your coworkers for more work.

> 
> Hi *supervisor*,
> 
> 
> I finished task X early and solved the issue by doing Y. I will be asking my coworkers for new tasks to do now, as we agreed upon before you left.
> 
> 
> Good luck with the business trip (or holiday, or whatever they're doing)
> 
> 
> Kind regards, mynameissecret
> 
> 
> 

This way, your supervisor knows what you're up to. If your coworkers fail to give you any new tasks, you can get back to your supervisor on this.

As for asking for new tasks from your coworkers, I would ask **in person first**. This way you have the best chance on getting a response fast. I like to start the conversation in person and if needed explain what my skills are. It might happen that they don't know of a task just by the top of their head. I then ask them to email me with any tasks they have available for me.

Now it can happen that they forget to send you a task, or maybe you just don't hear back from them. Send them an email at the end of the day asking for a task.

> 
> Hi coworker,
> 
> 
> Did you manage to find the time to assign some tasks to me? If not, do you know who else I could ask for a new task?
> 
> 
> Thanks for helping me,
> 
> 
> mynameissecret.
> 
> 
> 

Now, on to what you should be doing while **waiting for a new task**. As I understand, your supervisor did not give you any material to work on other than the first task. I would put time in learning new things. Read up on articles, search for tutorials etc. If you're unsure if you are allowed to do these things, you could include a line mentioning this in the email you initially send to your supervisor. This way you're letting him know you're doing something productive while waiting.