We are installing a Cooktop that calls for a 40amp circuit. I was given 6AWG wire (instead of the 8AWG wire I asked for) by mistake. The electrician saw the 6AWG wire (that I had already run) and used a 50amp breaker. The Cooktop installer requires that the breaker be 40amp. My electrician was concerned that using the larger 6AWG wire would lower the resistance therefore affecting the amps; he cited Ohm's law. He has refused to change the breaker from 50amps to 40amps for fear of damaging something or affecting safety. I thought the amp requirement was dictated by the device (it asks for 40amps) and the 40amp breaker would ensure that 40amps is all that it would get. Does anyone know of any dangers/issues if using a smaller breaker with 6AWG wire?
Your "electrician" is not one of the brighter bulbs in the pack.
The 40A is to protect the wiring and the device.
If the wiring is AT LEAST 8Ga then it's adequate to protect the wiring. It also protects 6Ga, (or 500 MCM for that matter) just fine, and it properly protects the device at the end of the wire just fine.
"Ohms law" has squat to do with this. You could have a cooktop located 3 feet from the breaker panel and connected with 8 Ga or one located 100 feet from the panel and connected with 6 Ga - the 8 Ga would have (much, about 20 times) lower resistance, because of the wire length. Upsizing wire for longer runs on heavy circuits is actually quite normal. As stated, not a particularly knowledgeable electrician you have there.
The breaker needs to be sized to protect the wire and the device.
Larger wire (which is a lower # due to the way wire sizes are named) can use a larger breaker. But a smaller breaker is always safe. 55A is the largest breaker you can normally use for 6 AWG copper. 40A is the largest breaker you can normally use for 8 AWG copper. But you can always use a smaller breaker - it will be 100% safe. That includes the very typical 50A (instead of 55A) for 6 AWG. But it can include lots of different things. For example, a 30A breaker on 8 AWG wire, a 15A breaker with 12 AWG wire (which can also use a 20A breaker), etc. You could even use a 15A breaker on 6 AWG wire - strange but nothing unsafe about it.
The device needs to be protected by an appropriate size breaker which is determined by the design of the device and is part of the UL (or equivalent) listing for the device. So if the cooktop calls for a 40A breaker then you must use a 40A breaker. You can't use a smaller breaker (probably safe, but you would get frequent nuisance trips which are inconvenient at best and lead to unsafe operation at worst if you (or a future owner) ends up "fixing" it later in an unsafe manner). And you can't use a larger breaker because the device is not rated for that - i.e., it expects to have the protection provided by a 40A breaker in order to handle any faults in a safe manner.
It is possible to have multiple valid breaker sizes. For example, a circuit consisting of 12 AWG wire and 15A duplex receptacles can use a 15A breaker (perfect match for the individual receptacles) or 20A breaker - OK because of the wire size (15A would only need 14 AWG) and a special exception for 20A circuits that allows for multiple 15A receptacles instead of 1-or-more 20A receptacles, and the 15A receptacles are designed to allow 20A passing through. Any normal plug-in 15A device can use a 20A receptacle. But that is not necessarily the case for 40A vs. 50A - and unless the cooktop instructions actually say it is OK to do so, you need to stick with 40A, even if the wire can handle 50A.
You're always allowed to upsize wire
What you have there is a 40A circuit, because it is breakered 40A per instructions.
On a 40A circuit you are allowed to use any cable 8 AWG or larger.
It's that simple.
6 AWG is larger than 8 AWG, so you are ducky-doo with the #6. Good call, since some better stoves/ranges want 50A or even 60A, and #6 is good for all that.
The only speedbump with the "any size or larger" is a very much larger wire may not physically fit on the breaker or panel lugs. In that case you need to simply pigtail to an intermediate size or metallurgy. For instance if you wisely chose 4 AWG Aluminum for your 400' long-run 30A dryer circuit, neither the 30A breaker nor socket will accept #4 nor aluminum. So you use Al-rated Polaris connectors to pigtail to #10 Cu, which will fit without trouble. #10Cu is good on a 30A circuit.
As for the electrician's "mistake" I don't see the problem. If he wasn't aware of the range specs, he made absolutely correct assumptions based on facts at hand. Many 40A ranges are dual-listed for 40 or 50A breakers, and both use the same socket. If wrong, it's a $9 change. No big.
The breaker now matches the wire rating (so you are protected against wall fires from overheating wires) and exceeds the cooktop rating (so it will not trigger accidentally). You are fine, but you'd have been fine with a 40A breaker as well. You state that the cooktop calls for a 40A breaker but that just means that the breaker (and wire) should be able to deliver at least 40A. The makers of the cooktop cannot delegate operating security to the wire breaker. Arguably without additional circuitry it would be a bad idea to mount the cooktop on a 5000A rail since it might melt into slag in case of a shortcircuit. The difference between 50A and 40A for the fault modes to be expected, however, are negligible. There just isn't a failure mode where the cooktop would continually draw, say, 48A through its internal wiring.
Unless, of course, some terminals have been mixed up and parts are running between different phases rather than between phase and ground, leading to a consistent too high load but not a short circuit. While the overall competence of your electrician does not seem overwhelming, his insistence on sticking with what he knows makes that unlikely. If you find that some of your plates heat water significantly faster than expected, you probably should have that double-checked.