A copper transmission cable 100100 km long and 10.010.0 cm in diameter carries a current of 125125 A. How much electrical energy is dissipated as thermal energy every hour?

Ch 25: Current, Resistance, and EMF

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 25: Current, Resistance, and EMF

Problem 25b

Chapter 25, Problem 25b

A copper transmission cable $100$ km long and $10.0$ cm in diameter carries a current of $125$ A. How much electrical energy is dissipated as thermal energy every hour?

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First, calculate the resistance of the copper cable using the formula for resistance: $ R = \frac{\rho L}{A} $, where $ \rho $ is the resistivity of copper, $ L $ is the length of the cable, and $ A $ is the cross-sectional area. The resistivity of copper is approximately $ 1.68 \times 10^{-8} \ \Omega \cdot m $.

Determine the cross-sectional area $ A $ of the cable using the formula for the area of a circle: $ A = \pi r^2 $, where $ r $ is the radius of the cable. Given the diameter is 10.0 cm, convert it to meters and find the radius.

Substitute the values of $ \rho $, $ L $, and $ A $ into the resistance formula to find the resistance $ R $ of the cable.

Use the power dissipation formula $ P = I^2 R $ to calculate the power dissipated as thermal energy, where $ I $ is the current flowing through the cable.

Finally, calculate the total energy dissipated over one hour by multiplying the power $ P $ by the time in seconds (1 hour = 3600 seconds) using the formula $ E = P \times t $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Electrical Resistance

Electrical resistance is a measure of the opposition to the flow of electric current through a conductor. It depends on the material's resistivity, length, and cross-sectional area. For a copper cable, resistance can be calculated using the formula R = ρL/A, where ρ is the resistivity, L is the length, and A is the cross-sectional area.

Joule's Law

Joule's Law relates the power dissipated in a resistor to the current flowing through it and its resistance. It is given by the formula P = I²R, where P is the power, I is the current, and R is the resistance. This law helps determine the thermal energy produced due to the resistance in the cable.

Energy Dissipation

Energy dissipation in electrical systems refers to the conversion of electrical energy into thermal energy due to resistance. The energy dissipated over time can be calculated using the formula E = Pt, where E is the energy, P is the power, and t is the time. This concept is crucial for understanding how much energy is lost as heat in the cable over a given period.

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