Current flow through resistor and insufficient source
If the source acts like a bench power supply and actively limits the current to 0.5A you will have 5V across the resistor. 

If the source acts like an ideal voltage source with a 20\$\Omega\$ resistor in series (10V open circuit, 0.5A into a short), the voltage across the external 10 ohm resistor will be less and thus the current will be (1/3) less than the above case.

There are other possibilities- batteries would behave somewhat differently. 

If the 0.5A limit is because of a fuse the current will be zero. 

If the limit is because it's an auto retry switching supply with current limiting the current will be short pulses and probably average fairly low.