To calculate the power loss in the circuit, we need to first determine the voltage drop due to the resistance of the #10 copper wire over the 120-foot distance. The power loss can then be calculated using Ohm's law and the formula for electrical power.
Step 1: Find the Resistance of the Wire
The resistance of a copper wire can be determined using the formula:
R=resistance per unit length×lengthR = \text{resistance per unit length} \times \text{length}R=resistance per unit length×length
For #10 AWG copper wire, the resistance is approximately
1.018 ohms per 1000 feet.
The total length of the wire is 120 feet (one way), so for a round trip, it is 240 feet.
Now, calculate the resistance:
R=(1.018 Ω1000 feet)×240 feet=0.2443 ΩR = \left( \frac{1.018 \, \Omega}{1000 \, \text{feet}} \right) \times 240 \, \text{feet} = 0.2443 \, \OmegaR=(1000feet1.018Ω)×240feet=0.2443Ω
Step 2: Calculate the Voltage Drop
The voltage drop VdropV_{\text{drop}}Vdrop across the wire can be calculated using Ohm's law:
Vdrop=I×RV_{\text{drop}} = I \times RVdrop=I×R
where:
- I=20 ampsI = 20 \, \text{amps}I=20amps
- R=0.2443 ΩR = 0.2443 \, \OmegaR=0.2443Ω
Substitute the values:
Vdrop=20 amps×0.2443 Ω=4.886 voltsV_{\text{drop}} = 20 \, \text{amps} \times 0.2443 \, \Omega = 4.886 \, \text{volts}Vdrop=20amps×0.2443Ω=4.886volts
Step 3: Calculate the Power Loss
The power loss PlossP_{\text{loss}}Ploss in the wire is due to the resistance and can be calculated as:
Ploss=I2×RP_{\text{loss}} = I^2 \times RPloss=I2×R
Substitute the values:
Ploss=(20 amps)2×0.2443 Ω=400×0.2443=97.72 wattsP_{\text{loss}} = (20 \, \text{amps})^2 \times 0.2443 \, \Omega = 400 \times 0.2443 = 97.72 \, \text{watts}Ploss=(20amps)2×0.2443Ω=400×0.2443=97.72watts
Final Answer
The power loss in the circuit is
97.72 watts.
