Question

A modern house is wired for \(115 \mathrm{~V}\), and the current is limited by circuit breakers to a maximum of \(200 .\) A. (For the purpose of this problem, treat these as DC quantities.) a) Calculate the minimum total resistance the circuitry in the house can have at any time. b) Calculate the maximum electrical power the house can consume.

Short answer

Solution: a) Calculate the minimum total resistance: R = V / I R = (115 V) / (200 A) R_min = 0.575 ohms b) Calculate the maximum electrical power: P = V * I P_max = (115 V) * (200 A) P_max = 23,000 W or 23 kW Answer: The minimum total resistance is 0.575 ohms, and the maximum electrical power the house can consume is 23,000 W or 23 kW.

Step-by-step solution

a) Calculate the minimum total resistance

Using Ohm's law, we have V = I * R, where V is voltage, I is current, and R is resistance. We need to find R when V = 115 V and I = 200 A. To get R, we rewrite Ohm's law as: R = V / I Now, substituting the values V = 115 V and I = 200 A, we get: R = (115 \mathrm{~V})/(200\mathrm{~A}) Calculating this, we get the minimum total resistance the circuitry in the house can have at any time: R_min.

b) Calculate the maximum electrical power

To calculate the maximum electrical power, we use the power equation: P = V * I Using the same values of V and I as in part a (V = 115 V and I = 200 A), we have: P_max = (115 \mathrm{~V})(200\mathrm{~A}) Calculating this, we get the maximum electrical power the house can consume: P_max.

Key concepts

Electrical Resistance

Electrical resistance is a measure of how much a material opposes the flow of electric current. It's often compared to water flowing through a pipe; just as narrower or longer pipes create more resistance to water flow, certain materials can hinder electrical current more than others.

Ohm's law, which states that voltage (\textbf{V}) equals current (\textbf{I}) times resistance (\textbf{R}), gives us the formula to calculate resistance: \[ R = \frac{V}{I} \].
In the case of the household wiring example, a low resistance allows more current to flow. However, too low resistance can be dangerous because it could result in overheating and potentially start a fire. That's why houses have circuit breakers, to prevent currents becoming too high. More on that in it's own section!

Electrical Power

Electrical power, measured in watts (\textbf{W}), is the rate at which electrical energy is consumed or converted into another form of energy, like heat or light. The formula for electric power is given by \[ P = V \times I \], where \textbf{P} is the power, \textbf{V} is the voltage, and \textbf{I} is the current.

In our household example, the circuit breakers limit the current, ensuring that the power we use stays within safe limits to prevent electrical fires or damage to the electrical system and appliances. Understanding the concept of power is fundamental when designing or modifying electrical systems in homes to ensure safety and energy efficiency.

Circuit Breakers

Circuit breakers are crucial components in household electrical systems designed to prevent overloading, which could lead to electrical fires. These devices automatically stop the electrical flow when the current exceeds a safe level.

The current rating of a circuit breaker, like the given 200 A in our household example, is the maximum current it can handle before it trips off. Calculating the resistance and power helps in determining the appropriate circuit breaker rating for a system. Properly rated and functioning circuit breakers are essential to any safe electrical setup, working as a guard against potential electric hazards.