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A power transmission line 50km long has a total resistance of 0.60 ohm. A generator produces 7 kilowatts at 100V. In order to reduce the power loss due to heating of the transmission line, the voltage is stepped up to 100kV using a transformer.

What percentage of the original power would be lost if the transformer were not used?

What percentage of the original power is lost when the transformer is used?

How would I figure this out? Any help is highly appreciated thanks!

That is some kinda tricky but we are gonna work it out
Thermal power consumed by the transmission line due to its resistance:
P(c)=RI^2
Where P(c) is power consumed
I= produced power divided by produced voltage: I=P/V
(1) In the case before voltage was raised
I=P/V=7000/100=70 A
Thermal power consumed in this case:
P(c)=RI^2=0.6*(70)^2=2940 Watt
Lost power percentage=2940/7000= 42.0%

(2) In the case after voltage has been raised:
P=IV(produced power)
When P is constant then we have
IV=constant ⇒ I1V1=I2V2 ⇒ I2/I1=V1/V2
I2/I1=100/100K=1/1000

And since the lost power percentage is directly proportional to (I^2)
then the ratio between the lost power after and before the voltage raising is equal to (I2/I1)^2 =(1/1000)^2=1/10^6

In math language: P(c)2/P(c)1=(I2/I1)^2=1/10^6
This implies that: P(c)2=P(c)1/10^6
then lost power percentage after voltage raising = 42/10^6 = 42*10^-6%

The second part can be obtained the same way the first part was done as follows:

I=P/V=7000/100000=0.07A
P(c)=RI^2=0.6*(0.07)^2= 0.00294 watt

Lost power percentage in this case=0.00294/7000=42*10^-6%

Comments:
1- This problem would be a little bit more complicated but even more realistic if the efficiency of the transformer was considered.
2- One should differentiate between Power produced and Power consumed (big difference).

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