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Question:

Does increasing the thickness of electrical wiring in a home lower energy, meter readings and bills?

I heard someone at an energy conference refer to a study that showed that going to a one size thicker wire in a home -- I am assuming on the American wire gauge, AWG, system -- will lead to a significant drop in home electrical energy usage. I write significant because the percentage was so big I can't remember it exactly.

Answer:

Depends on how many amps you are using in the first place, I assume. If you underestimated the amount of current usage in a particular circuit in the first place, and you undersized the wire as a result, then I suppose that the smaller wire will have more apparent resistance due to it being unable to handle the current load without over hearting. They seem to be suggesting that the standard gage wire used in American homes is undersized because our standards are off and the electrical code is wrong.
In the case of a light bulb bigger wire will cause the same light bulb to use more power. Since the voltage drop to the light bulb would be greater with smaller wire, hence voltage at the bulb would be lower. PowerV^2/R would be less for a light bulb using smaller wire. The bulb would be dimmer with less voltage to it. With a heavily loaded motor the results would be different due to the effect of reduced voltage on the speed torque curve. I think the people who suggest that this would save energy are in the business of selling copper!
As already noted, these are available. They are sometimes known as Meter Socket Blank Covers. Most are plastic these days. Old ones were glass. The meter socket is a standardized dimension. No need to measure. Very unlikely to be found at big box stores. The power company may not be a bad idea (may even get it for free). The other likely place is going to be at an electrical supply store (Some names that may be in your area are: Platt, All-Phase, Consolidated Electrical Distributors/CED, Stusser, Graybar). Of course, the suggestion to remove it and repair the hole was not bad either. I have also seen several other answers from Mr. 50 yrs retired electrician. I'm not sure where he did his 50 yrs. Not likely anywhere that the NEC was enforced. If I'm wrong, I should move and start finding his old customers. I could spend the rest of my career fixing work like that.
Theoretically yes by increasing the wire gauge there will be lower I?r losses (Watts) as the resistance r is lower in wire of greater diameter. HOWEVER, the resistance value of copper wires of just a couple of gauges difference is so small as to not contribute to any difference in the meter reading/monthly bill. Considering that an 8 gauge wire has a resistance of 8 ohms per mile and a 12 gauge wire's resistance is 3 ohms per mile, the difference being 5 ohms per mile. Say you had 1 mile of wire in your house (which is impossible) with 10A running constantly. The wire heating loss would be 10? x 5 500W or 0.5kW. To have this for 1 hour would consume 0.5kWhrs, at say 10 cents/h 5 cents. Now, there would be not more than 100 feet loop length on a branch circuit so the cost would be 100/5280 x 5cents 0.01cts/h. Conclusion: Your energy conference leader is talking RUBBISH.

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