A typical cost for electrical power in Europe is € 0.219 per...

A typical cost for electrical power in Europe is $€ 0.219$ per kilowatt-hour. (a) Some people leave their porch light on all the time. What is the yearly cost to keep a $75 \mathrm{~W}$ bulb burning day and night? (b) Suppose your refrigerator uses $200 \mathrm{~W}$ of power when it's running, and it runs 8 hours a day. What is the yearly cost of operating your refrigerator? 25.66 -o A cylindrical copper cable $1.50 \mathrm{~km}$ long is connected across a $230.0 \mathrm{~V}$ potential difference. (a) What should be its diameter so that it produces heat at a rate of $90.0 \mathrm{~W} ?$ (b) What is the electric field inside the cable under these conditions?

Key Concepts

Electric Field in a Conductor

The electric field inside a conductor under a potential difference is a measure of the force per unit charge experienced by charges within the material. It is typically calculated as the potential difference divided by the length of the conductor. This concept is important for understanding how voltage is distributed along a conductor and for analyzing the behavior of current flow under electrical load.

Conductor Geometry

This concept involves the mathematical relationship between the physical dimensions of a conductor and its electrical properties, particularly resistance. For a cylindrical conductor, the cross-sectional area is determined by its diameter, and it plays a key role in the resistance calculation. Grasping this relationship is important for designing conductors to meet specific electrical performance criteria.

Resistivity and Resistance

Resistivity is an intrinsic property of materials that quantifies how strongly they resist the flow of electric current. The resistance of a conductor is determined by its resistivity, length, and cross-sectional area. The concept is pivotal when designing or analyzing components like cables where electrical conduction and heat production are directly affected by these properties.

Joule Heating and Power Dissipation

Joule heating, or resistive heating, describes the process by which electrical energy is converted into heat in a conductor due to its resistance. This is directly related to the power dissipated in the material, which can be calculated using the formula P = V²/R, where V is the voltage across the conductor and R is its resistance. Understanding this is critical in applications where controlling heat production is necessary.

Watt-Hour Conversion

A crucial aspect of these problems is converting power ratings, given in watts, into energy measured in watt-hours or kilowatt-hours. This involves multiplying the power of the device by the number of operating hours. This conversion is fundamental for quantifying energy use over time and for subsequently determining costs or load requirements in electrical systems.

Cost Calculation

This concept describes how to determine the operational expense of using an electrical device based on its energy consumption and the rate charged per unit of energy. By converting energy usage into the appropriate units (typically kilowatt-hours) and multiplying by the cost per unit, one can derive the monetary cost of operating a device over a given period.

Energy Consumption

This concept involves calculating the total amount of energy used by an electrical device over a period of time. It is based on the relationship between power (measured in watts) and time (often converted into hours), leading to energy units such as kilowatt-hours. This conversion is essential for cost calculation applications as well as understanding the performance and load on electrical systems.

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