A generating station is producing 2.00 106 W of power that is to be sent to a small town located 7.0 km away. Each of the two wires that comprise the transmission line has a resistance per kilometer of length of 5.00 10-2 /km.
(a) Find the power lost in heating the wires if the power is transmitted at 1300 V.
W
(b) A 120:1 step-up transformer is used to raise the voltage before the power is transmitted. How much power is now lost in heating the wires?
W
I did this yesterday for someone:
Power = V i = 1300 * i = 2*10^6
so
i = (2/1.3)*10^3 = 1.54 * 10^3 amps (big current!)
R = 2 * 7 * 5 * 10^-2 = .7 Ohms
Power loss = i^2 R = (1.54)^2*10^6*.7 = 1.66*10^6 Watts (we lose about as much as we use)
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Now use a high voltage line
i = 1.54*10^3/120 = 12.8 amps (reasonable now)
power loss = i^2 R = 12.8^2 (.7) = 115 Watts, about a 100 watt light bulb worth lost now.
(a) To find the power lost in heating the wires, we need to calculate the total resistance of the transmission line and then use the formula P = I^2*R, where P is the power lost, I is the current, and R is the resistance.
Step 1: Calculate the total resistance of the transmission line.
Given: Resistance per kilometer of length = 5.00 * 10^-2 /km
Distance = 7.0 km
Total resistance = Resistance per kilometer of length * Distance
Total resistance = (5.00 * 10^-2 /km) * (7.0 km)
Total resistance = 0.35 Ω
Step 2: Calculate the current flowing through the transmission line.
Given: Power = 2.00 * 10^6 W
Voltage = 1300 V
Using the formula P = IV, we can solve for current:
2.00 * 10^6 W = 1300 V * I
I = (2.00 * 10^6) / 1300
I = 1538.46 A
Step 3: Calculate the power lost in heating the wires.
Using the formula P = I^2*R:
P = (1538.46 A)^2 * 0.35 Ω
P ≈ 857,142 W
Therefore, the power lost in heating the wires is approximately 857,142 W.
(b) When a 120:1 step-up transformer is used, the voltage is increased by a factor of 120. Therefore, the new voltage is 120 * 1300 V = 156,000 V.
To calculate the power lost in heating the wires with the new voltage, we follow the same steps as in part (a), but using the new voltage.
Step 1: Total resistance remains the same as in part (a).
Step 2: Calculate the current:
Power = 2.00 * 10^6 W
Voltage = 156,000 V
I = Power / Voltage
I = (2.00 * 10^6) / 156,000
I = 12.82 A
Step 3: Calculate the power lost in heating the wires:
P = I^2 * R
P = (12.82 A)^2 * 0.35 Ω
P ≈ 58.9 W
Therefore, when using a 120:1 step-up transformer, the power lost in heating the wires is approximately 58.9 W.