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
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.
To find the power lost in heating the wires, we can use the formula for power loss due to resistance:
Power Loss = (Resistance per unit length) * (Length) * (Current)^2
(a) First, let's find the current flowing through the transmission line. We can use Ohm's Law:
V = I * R
Where V is the voltage and R is the resistance. Rearranging the formula, we can solve for I:
I = V / R
Given that the voltage is 1300 V and the resistance per kilometer is 5.00 * 10^-2 /km, we can calculate the current:
I = 1300 V / (5.00 * 10^-2 /km) = 2.60 * 10^4 A
Next, we need to calculate the total resistance in the transmission line. Since the transmission line has a total length of 7.0 km, the total resistance can be found by multiplying the resistance per kilometer by the length:
Total Resistance = (Resistance per kilometer) * (Length) = (5.00 * 10^-2 /km) * (7.0 km) = 3.50 * 10^-1
Finally, substitute the values into the formula for power loss:
Power Loss = (Resistance per unit length) * (Length) * (Current)^2 = (3.50 * 10^-1 Ω) * (7.0 km) * (2.60 * 10^4 A)^2 = 1.89 * 10^11 W
Therefore, the power lost in heating the wires is 1.89 * 10^11 W.
(b) If a 120:1 step-up transformer is used to raise the voltage, the new voltage will be:
New Voltage = (Transformer ratio) * (Original voltage) = 120 * 1300 V = 1.56 * 10^5 V
Using the new voltage, we can calculate the new current:
New Current = New Voltage / Resistance = (1.56 * 10^5 V) / (5.00 * 10^-2 /km) = 3.12 * 10^6 A
Now, recalculate the total resistance using the same formula as in part (a):
Total Resistance = (Resistance per kilometer) * (Length) = (5.00 * 10^-2 /km) * (7.0 km) = 3.50 * 10^-1 Ω
Finally, substitute the new values into the formula for power loss:
Power Loss = (Resistance per unit length) * (Length) * (New Current)^2 = (3.50 * 10^-1 Ω) * (7.0 km) * (3.12 * 10^6 A)^2 = 6.47 * 10^14 W
Therefore, the power lost in heating the wires after using the step-up transformer is 6.47 * 10^14 W.