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Chapter 7: Problem 1

Would electric potential energy be meaningful if the electric field were not conservative?

Short Answer

Expert verified
No, electric potential energy wouldn't be meaningful if the electric field were not conservative because potential energy relies on a force being conservative.

Step by step solution

01

Understand conservative forces

A conservative force is a force with the property that the work done in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the total work done is zero.
02

Explain potential energy

Potential energy is the energy held by an object because of its position relative to other objects. In the context of electricity, electric potential energy is associated with the configuration of charged particles. The work done in moving a charge in an electric field (force field) gives rise to potential energy, similarly to lifting an object against gravitational pull.
03

Relate conservative force with potential energy

A force field is said to be conservative if it has a potential energy associated with it. Therefore, potential energy (including electric potential energy) is meaningful only for conservative forces because it relies on the notion that the work done is path independent.
04

Conclude the exercise

So, if the electric field were not conservative, it would not make sense to speak of electric potential energy.

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Key Concepts

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

Electric Potential Energy
Electric potential energy refers to the energy that a charge possesses due to its position in an electric field. The concept parallels gravitational potential energy, where height determines energy in a gravitational field. Similarly, in an electric field, a charge's position relative to other charges determines its electric potential energy.
  • Potential energy increases as a charge is moved against the direction of the electric field.
  • This is because work needs to be performed to move the charge, which stores energy.
  • Conversely, moving with the field decreases potential energy.
This energy is crucial in understanding electric forces and their effects on charge movement. It forms the basis for various applications in technology and science, such as how batteries function by storing and transferring energy.
Path Independence
Path independence is a vital concept in understanding conservative forces within force fields, such as electric fields. It means that the work done by or against a conservative force as a particle moves from one point to another is independent of the specific route taken. Unlike non-conservative forces, which depend on the path taken, conservative forces simplify the computation of work.
  • For example, the work done in an electric field relies solely on the start and end points, not on how a charge gets there.
  • This quality allows for the definition of potential energy, which counts only initial and final positions.
  • In any closed loop, a charge moved in a conservative field would result in zero net work done.
This principle helps in solving complex problems by reducing the need for integrating paths, and is central to the existence of potential energy.
Work Done in Force Fields
Understanding work done in force fields, like electric fields, is crucial in physics exercises. Work done relates to energy transferred when a force moves an object. In an electric field, calculating this involves considering the electric force exerted on a charge and the distance over which it acts.
  • The formula for work done, when a conservative force is involved, is: \(W = -\Delta U\), where \(W\) is the work done, and \(\Delta U\) is the change in potential energy.
  • A conservative field allows figuring the work easily since it's just about potential energy differences, not the specific journey.
  • For non-conservative fields, calculating work becomes complicated, since the potential energy concept is not applicable.
Recognizing how work interacts with potential energy sheds light on the behavior of charges and helps predict movement and energy conservation in diverse systems.

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Most popular questions from this chapter

Use the electric field of a finite sphere with constant volume charge density to calculate the electric potential, throughout space. Then check your results by calculating the electric field from the potential.

(a) What is the direction and magnitude of an electric field that supports the weight of a free electron near the surface of Earth? (b) Discuss what the small value for this field implies regarding the relative strength of the gravitational and electrostatic forces.

Two parallel plates \(10 \mathrm{cm}\) on a side are given equal and opposite charges of magnitude \(5.0 \times 10^{-9} \mathrm{C}\). The plates are \(1.5 \mathrm{mm}\) apart. What is the potential difference between the plates?

Suppose you have a map of equipotential surfaces spaced 1.0 V apart. What do the distances between the surfaces in a particular region tell you about the strength of the \(\overrightarrow{\mathbf{E}}\) in that region?

In a Geiger counter, a thin metallic wire at the center of a metallic tube is kept at a high voltage with respect to the metal tube. Ionizing radiation entering the tube knocks electrons off gas molecules or sides of the tube that then accelerate towards the center wire, knocking off even more electrons. This process eventually leads to an avalanche that is detectable as a current. A particular Geiger counter has a tube of radius \(R\) and the inner wire of radius \(a\) is at a potential of \(V_{0}\) volts with respect to the outer metal tube. Consider a point \(P\) at a distance \(s\) from the center wire and far away from the ends. (a) Find a formula for the electric field at a point \(P\) inside using the infinite wire approximation. (b) Find a formula for the electric potential at a point P inside. (c) Use \(V_{0}=900 \mathrm{V}, a=3.00 \mathrm{mm}, R=2.00 \mathrm{cm}, \quad\) and find the value of the electric field at a point \(1.00 \mathrm{cm}\) from the center.
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