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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.

Wire

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.

Device

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.

If you ever change to an appliance that wants 50A, you have a 50A breaker. Ydou are allowed any wire 6 AWG or larger.