Recently I have been learning about the primary differences between AC and DC current and, as an extrapolation, the reasons for where they are used in technology. One of the primary benefits given for AC power in power transmission is that through the use of transformers we can step-up the voltage (and by the same token step down the current) so that less energy is lost due to Ohmic heating. In doing this, given the formula for power being, $P = I^2R$ , and so the formula for Ohmic heating becomes $P = I^2R$.

Where my confusion lies, then, is in the fact that this can be re-written as $$P = V^2/R$$ As we step down the current to lower ohmic heating, we at the same time are stepping up the voltage, which the above equation seems to imply will have the same effect as not having changed it at all. The way I see this as working is that given the circuit is in series (imagining a very simplified model of a city, the power grid and transmission wires), current will be constant throughout the circuit, regardless of where we look, whereas potential difference will not. However I'm having a little trouble visualizing this - why exactly this inconsistency vs consistency will effect the results for Ohmic heating in the transmission wires - that is, why does stepping up the voltage change anything at all if the formula can also be written as $P = V^2/R $ ? Would this mean that circuits connected in parallel would experience different effects - If anyone could help clarify this that would be great. I found this question a little hard to phrase so if it needs further clarification please let me know.

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    $\begingroup$ This has been asked and answered here already. You're not thinking clearly about what the voltage $V$ in your third equation represents. If $P$ is the power due to Ohmic heating, then the voltage $V$ in $P=V^2/R$ is not the stepped up voltage of the source but rather, the voltage drop along the length of the wire which is proportional to the current through: $$V_R = I\cdot R,\,P_R =\frac{V^2_R}{R} = I^2\cdot R$$ $\endgroup$ – Alfred Centauri Mar 28 '18 at 12:17

You can only re-write the formula if Ohm's $U = RI$ law is applicable. This means that current and voltage are proportional. $$ P \propto I^2 \propto U^2$$ Consequently, the following statement does not make sense:

As we step down the current to lower ohmic heating, we at the same time are stepping up the voltage, which the above equation seems to imply

Where does the equation imply this?

Concerning your statement that AC voltage helps to reduce energy loss, I'd like to quote Wikipedia and references therin: "Usually, a transformer is placed between the lines and consumption. When a high-voltage, low-intensity current in the primary circuit (before the transformer) is converted into a low-voltage, high-intensity current in the secondary circuit (after the transformer), the equivalent resistance of the secondary circuit becomes higher and transmission losses are reduced in proportion. During the War of Currents, AC installations could use transformers to reduce line losses by Joule heating, at the cost of higher voltage in the transmission lines, compared to DC installations." How this works is explained here: Advantage of high-voltage power transmission.


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