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Chapter 10: Problem 6

A resistor has a voltage difference \(\Delta V\) across it, causing a current \(I\) to flow. (a) Find an equation for the power it dissipates as heat in terms of the variables \(I\) and \(R\) only, eliminating \(\Delta V\). (answer check available at lightandmatter.com) (b) If an electrical line coming to your house is to carry a given amount of current, interpret your equation from part a to explain whether the wire's resistance should be small, or large.

Short Answer

Expert verified
(a) Power is \( P = I^2 \cdot R \). (b) The wire's resistance should be small to decrease heat dissipation.

Step by step solution

01

Understand Power Formula

The electrical power dissipated by a resistor can be calculated using the formula \( P = \Delta V \cdot I \), where \( P \) is power, \( \Delta V \) is voltage, and \( I \) is current.
02

Apply Ohm's Law

Ohm's Law is \( \Delta V = I \cdot R \), where \( R \) is resistance. This allows us to express the voltage difference \( \Delta V \) in terms of current \( I \) and resistance \( R \).
03

Substitute Ohm's Law into Power Formula

Using the expression from Ohm's Law, substitute \( \Delta V = I \cdot R \) into the power formula: \[ P = (I \cdot R) \cdot I = I^2 \cdot R \]. Thus, the power \( P \) can be expressed in terms of current \( I \) and resistance \( R \) as \( P = I^2 \cdot R \).
04

Interpret the Final Equation

The equation \( P = I^2 \cdot R \) shows that the power dissipated as heat is directly proportional to the resistance \( R \). This implies that for a given current \( I \), increasing the resistance \( R \) will increase the power dissipated as heat.
05

Analyze Resistance Requirement for Electrical Line

To minimize power loss as heat in an electrical line carrying a given current, the resistance \( R \) should be as small as possible. This reduces the power \( P = I^2 \cdot R \) dissipated as heat in the wires, making the transmission more efficient.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ohm's Law
Ohm's Law is a fundamental principle in electronics, relating voltage (6 V), current (76 I), and resistance (76 R) in an electrical circuit. It states that the voltage difference across a resistor is directly proportional to the current flowing through it. Mathematically, Ohm's Law is expressed as 76 V = I 76 R. This simple formula is extremely useful, allowing you to determine any one of the three variables if the other two are known.

In practical terms, if you know how much current is flowing and the resistance of the element, you can easily find out how much potential difference (voltage) is required to maintain that flow. Conversely, you can determine the current flow if you know the voltage across and the resistance of the component. This makes Ohm's Law indispensable in both designing and analyzing circuits.

One interesting aspect of Ohm's Law is its implication for power dissipation in resistors. As we will explore, this law is crucial in figuring out how much power is turned into heat when current flows through a resistance.
Electrical Resistance
Electrical Resistance is a measure of how much an object opposes the flow of electric current. The unit of resistance is the ohm (惟), named after Georg Ohm. When electric current flows through a material, the resistance determines how much voltage is needed to push a certain amount of current through it. A higher resistance means more voltage is needed for the same amount of current.

Resistance in any material is determined by its nature, thickness, length, and temperature:
  • Materials like copper and silver, which have low resistance, are excellent conductors.
  • Thicker wires have less resistance than thinner ones.
  • Shorter wires offer less resistance compared to longer ones.
  • Most materials increase in resistance as their temperature rises.
When considering power dissipation, we see the role of resistance in the formula 76 P = I^2 R. Higher resistance leads to more power being dissipated as heat, for a constant current. That's why resistors heat up when electric current flows through them. This concept is especially crucial when designing energy-efficient electrical systems, like the ones used for power transmission.
Electric Current
Electric Current refers to the flow of electric charge in a conductor. It is generally carried by moving electrons through a wire and is measured in amperes (A). Understanding current is essential for grasping how electrical circuits work.

There are two types of electric current:
  • Direct Current (DC), where the electric charges flow in one direction.
  • Alternating Current (AC), where the charge flow reverses direction periodically.
The direction of current is a convention based on the flow of positive charges. However, in reality, it's often the negatively charged electrons that move.

Electric current can change depending on the resistance and voltage in the circuit. According to Ohm's Law, a higher voltage or lower resistance will result in a higher current. In practice, understanding how current works can help you determine how much power will be used or dissipated by a component in the circuit. This is vital when examining the efficiency of devices and systems, ensuring they work safely and effectively without overheating or energy loss.

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Most popular questions from this chapter

A silk thread is uniformly charged by rubbing it with llama fur. The thread is then dangled vertically above a metal plate and released. As each part of the thread makes contact with the conducting plate, its charge is deposited onto the plate. Since the thread is accelerating due to gravity, the rate of charge deposition increases with time, and by time \(t\) the cumulative amount of charge is \(q=c t^{2}\), where \(c\) is a constant. (a) Find the current flowing onto the plate.(answer check available at lightandmatter.com) (b) Suppose that the charge is immediately carried away through a resistance \(R .\) Find the power dissipated as heat.(answer check available at lightandmatter.com)

What resistance values can be created by combining a \(1 \mathrm{k} \Omega\) resistor and a \(10 \mathrm{k} \Omega\) resistor? (solution in the pdf version of the book)

You are given a battery, a flashlight bulb, and a single piece of wire. Draw at least two configurations of these items that would result in lighting up the bulb, and at least two that would not light it. (Don't draw schematics.) If you're not sure what's going on, borrow the materials from your instructor and try it. Note that the bulb has two electrical contacts: one is the threaded metal jacket, and the other is the tip (at the bottom in the figure). [Problem by Arnold Arons.]

How many different resistance values can be created by combining three unequal resistors? (Don't count possibilities in which not all the resistors are used, i.e., ones in which there is zero current in one or more of them.)

In AM (amplitude-modulated) radio, an audio signal \(f(t)\) is multiplied by a sine wave \(\sin \omega t\) in the megahertz frequency range. For simplicity, let's imagine that the transmitting antenna is a whip, and that charge goes back and forth between the top and bottom. Suppose that, during a certain time interval, the audio signal varies linearly with time, giving a charge \(q=(a+b t) \sin \omega t\) at the top of the whip and \(-q\) at the bottom. Find the current as a function of time. (answer check available at lightandmatter.com)
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