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Is there a UMC extension cord available anywhere?

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That is the part that is confusing that I missed. He pointed back to the transformer, but then he makes a disconnection in the wire that goes out to the bulb, so it looks like making a disconnection out in the branch circuit. That looks like the case where you snip one of the two wires of the power cord of a lamp. The lamp will not stay lit. That tiny little one inch loop of wire that you see on that block that the bulbs are screwed into is apparently the same area that would be the main circuit panel of a house or the RV with its power buses.

(I know that you realized what you missed but I want to help explain for others...)

The neutral, and its connection to the center of the transformer, is what keeps the voltage balanced between the two legs. The neutral only carries the imbalance of the current between the two legs of the split-phase service. If the 120V loads on both sides of the split-phase service (L1-N and N-L2), then the neutral between the transformer and the panel never carries current.

Yes, if you cut the neutral on a single 120V appliance, it doesn't work, simply because you have no circuit (no wire to return the current).

However, if you break the neutral from the transformer to the panel, you still have a path for current to return. That's because in a split-phase service, you have circuits on the "left" side of the split-phase (L1 to neutral) and you have circuits on the "right" side of the split phase (L2 to neutral). All neutrals are connected together. In effect, the "left" side and "right" side loads get connected in series.

Normally, when the neutral is present, the conductor connecting to the center of the transformer can return the current imbalance, keeping the voltage constant. But if that connection goes away to keep the voltage constant, then voltage drop in series-connected loads is in the same proportion to the resistance of the loads.

Let's say that the right side load is 23x that of the left-side load... You'll see 10 volts on one leg and 230 volts on the other.

Check out this powerpoint... it shows the effects pretty well:
http://code-elec.com/userimages/Lost%20Neutral.ppt

As you noted, there is a panel in the RV - but even in some appliances this happens. For example, in a range, you might find the control panel is connected from L1-N while the oven light is connected from N-L2. If the neutral from the home were not connected via the neutral on the 14-50 or 10-50, you'll find the same problem.
 
As you noted, there is a panel in the RV - but even in some appliances this happens. For example, in a range, you might find the control panel is connected from L1-N while the oven light is connected from N-L2. If the neutral from the home were not connected via the neutral on the 14-50 or 10-50, you'll find the same problem.

I had already started to wonder about that. In a 120/240 appliance like that, wouldn’t they build them to use just one hot side for the 120V, just for extra reliability in case the neutral did get broken or damaged? I see no pros and only cons from connecting up both separate 120V circuits in an appliance.
 
I had already started to wonder about that. In a 120/240 appliance like that, wouldn’t they build them to use just one hot side for the 120V, just for extra reliability in case the neutral did get broken or damaged? I see no pros and only cons from connecting up both separate 120V circuits in an appliance.

Most of them do, because the power drawn by control panels is negligible. But in others it's not the case - think about an oven that has a rotisserie motor, lights, control logic, etc. The goal would be to keep power somewhat balanced and the risk of a broken neutral is fairly minimal from the appliance designer's standpoint, and it's only one kind of neutral break that is defended against -- the appliance circuit itself.