Non-contact detector shows elec. from wall switch, but light will not come on
Ever use air tools?  The air pressure line comes from the compressor, goes to the tool, and where does the air go after that?  it just disappears, into the magical aether, or really, just the air.  

Ever use hydraulic tools?  The hydraulic line comes from the pump, through the tool, and then there's a second hydraulic line back to the pump. Why? So you don't flood the shop with hydraulic fluid! 

Electricity does not work like a compressor where the power comes out the hot and just disappears into the aether  (except in Australia, where it does, but nevermind that). 

Actually, electricity works like hydraulics, except it's two-way, so the return line is just as important as the supply.  That return line is where your problem lies. This is called the neutral wire. 

Measuring the neutral wire with  your voltage tester may be useful. It should read no voltage. If you measure hot voltage there (with the light on), the problem is between there and the breaker.