If Edison doubled the length of his delivery wires, while keeping the currents through them the same, what would happen to the power they consumed?

The relationship between resistance and the length of a wire is an important concept in electrical engineering. According to the formula for resistance, \(R = \rho \frac{L}{A}\), the resistance \(R\) of a wire depends on three factors: the resistivity \(\rho\) of the material, the length \(L\) of the wire, and its cross-sectional area \(A\).

When the length \(L\) of the wire is doubled, the resistance also doubles. This is because resistance is directly proportional to length.

For example, if a wire's original resistance is \(R\), and the length is doubled, the new resistance becomes \(2R\). Hence, if you have a 10-meter wire with 5 ohms resistance, extending it to 20 meters would increase the resistance to 10 ohms.

This principle is crucial in the design of electrical systems to prevent excessive power loss and ensure efficient energy delivery.

Ohm's Law is a fundamental principle in electrical engineering that relates voltage, current, and resistance in an electrical circuit. The law is represented by the equation \(V = IR\), where \(V\) is the voltage, \(I\) is the current, and \(R\) is the resistance.

This formula helps to understand how changes in voltage and resistance affect current flow. For instance, if you increase the voltage while keeping the resistance the same, the current will increase.

Similarly, if you increase the resistance while keeping the voltage the same, the current will decrease.

In practical applications, Ohm's Law allows engineers to predict how electrical circuits will behave under various conditions, aiding in the design and troubleshooting of electrical systems.

Power in electrical circuits represents the rate at which electrical energy is converted into another form, such as heat, light, or mechanical energy. The power consumed in a circuit can be calculated using different formulas based on Ohm's Law:

• \(P = VI\): Power (\(P\)) equals voltage (\(V\)) times current (\(I\)).
• \(P = I^2 R\): Power equals the square of the current times the resistance.
• \(P = \frac{V^2}{R}\): Power equals the square of the voltage divided by the resistance.

In the given example, the power consumed by the wires is described by \(P = I^2 R\). When the length of the wire is doubled, its resistance also doubles, resulting in doubled power consumption.

So, if the original power consumption was \(P = I^2 R\), then after doubling the length (and thus the resistance), the new power is \(P_{new} = I^2 (2R) = 2I^2 R\). This means the power consumption also doubles.

Understanding these relationships helps manage energy use and efficiency in electrical systems, preventing unnecessary power loss and ensuring operational stability.