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2.7.3 - Potential energy of a dipole in an external field

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Understanding Dipoles and Torque

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Teacher
Teacher

Today, we will explore how a dipole behaves in a uniform electric field. What do you think happens to a dipole composed of two equal but opposite charges when placed in such a field?

Student 1
Student 1

I think it will just stay in one position because the opposite charges might counterbalance each other.

Teacher
Teacher

Good thought, but actually, while the dipole does not experience a net force, it does experience torque! The torque is given by the cross product of the dipole moment p and the electric field E. This means the dipole wants to align itself with the field.

Student 2
Student 2

What does that mean for the dipole's position?

Teacher
Teacher

It means it will try to rotate to minimize its potential energy within the field!

Work Done by External Torque

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Teacher
Teacher

Now, when we apply an external torque to our dipole, it can counteract the torque from the electric field. If the dipole rotates from an angle θ0 to θ1 at a constant angular speed, we can calculate the work done by that external torque. Can anyone express that relationship?

Student 3
Student 3

Is it related to the potential energy somehow?

Teacher
Teacher

Exactly! The work done can be expressed as W = pE (cos θ - cos θ0). This work contributes to potential energy U(q) that changes with the dipole's orientation relative to the electric field.

Student 4
Student 4

What about the zero reference for potential energy?

Teacher
Teacher

Great question! We typically set our reference point at θ0 = π/2, which simplifies calculations since it helps us understand how energy changes with angle.

Potential Energy Calculation Example

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Teacher
Teacher

Let’s look at a practical example. Suppose we have a molecule with a dipole moment of 10–29 C m in a strong electric field of 106 V/m. If the field's direction changes by 60 degrees, what can we infer about the energy involved?

Student 1
Student 1

We need to find the initial and final potential energy to determine how much energy is released, right?

Teacher
Teacher

Exactly! By calculating the initial and final potential energy, we can demonstrate that the change in potential energy indicates the energy released as heat during the process.

Student 2
Student 2

So, the dipole's alignment impacts energy states significantly?

Teacher
Teacher

Absolutely! The dipole's changes in alignment reflect how energy is absorbed or released, which is crucial in molecular interactions.

Introduction & Overview

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Quick Overview

This section discusses the potential energy of a dipole placed in a uniform electric field, detailing the associated torque and work done.

Standard

It defines the potential energy of a dipole in a uniform electric field, explaining the relations between torque, work done by external forces, and the concept of potential energy. A detailed example illustrates the heat released in a real-world scenario when a dipole is aligned with the electric field.

Detailed

In this section, we consider a dipole composed of charges +q and -q, placed within a uniform electric field E. While the dipole does not experience a net force, it does experience a torque given by t = p × E, attempting to align the dipole with the electric field. An external torque can be applied to counter this effect, allowing us to calculate the work done, expressed by the formula W = pE (cos θ - cos θ0). This work accumulates as potential energy U(q), where a natural zero reference is chosen at an inclination of θ0 = π/2. Additionally, potential energy can be explored from the perspective of each charge’s location in the electric field. A practical example illustrates how changing the electric field's direction affects the energy dynamics of a substance's dipoles.

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Audio Book

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Understanding the Electric Dipole in a Uniform Electric Field

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Consider a dipole with charges q = +q and q = –q placed in a uniform electric field E. As seen in the last chapter, in a uniform electric field, the dipole experiences no net force but experiences a torque τ given by τ = p × E, which will tend to rotate it (unless p is parallel or antiparallel to E).

Detailed Explanation

In this chunk, we are looking at an electric dipole placed in a uniform electric field. An electric dipole consists of two equal and opposite charges separated by a distance. When this dipole is placed in an external electric field, it feels no net force because the forces exerted by the electric field on the two charges are equal in magnitude and opposite in direction, thus canceling each other. However, it experiences a torque (τ) given by the vector product of the dipole moment (p) and the electric field (E). This torque tends to rotate the dipole either to align it with the field or against it, depending on its initial orientation. If the dipole is aligned with the field, the torque is zero.

Examples & Analogies

Think of a windmill. The wind hits the blades of the windmill on one side and applies force, rotating the windmill. In a similar way, an electric dipole in a field feels a 'wind' of electric force that makes it want to turn toward the direction that reduces its potential energy.

Calculating Work Done by External Torque

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Suppose an external torque τ_ext is applied to just neutralize the torque τ and rotates it from an initial angle θ₀ to an angle θ at an infinitesimal angular speed without angular acceleration. The work done by the external torque can be expressed as W = pE cos(θ₀) - pE cos(θ).

Detailed Explanation

In this portion, we learn about the work done by an external torque during the rotation of a dipole in the electric field. When an external torque is applied to counteract the torque caused by the electric field, it assists the dipole in rotating. The work done here can be calculated as the difference in the potential energy of the dipole at the two angles (θ₀ and θ). The expression involves the dipole moment (p) and the electric field (E), showing how the work done relates to the angle of rotation and the energy associated with the dipole's orientation relative to the field.

Examples & Analogies

Imagine trying to adjust the angle of a photo frame on your wall. If the frame is pushed slightly out of position due to an external effect (like wind), you would have to exert energy to push it back to align perfectly with your desired angle. That energy you use is akin to the work done against the electric torque acting on the dipole.

Potential Energy of the Dipole

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The work done against the external field is stored as potential energy U(θ) associated with the orientation of dipole in the electric field. It can be expressed as U(θ) = -pE cos(θ).

Detailed Explanation

In this chunk, we focus on the concept of potential energy as it relates to the orientation of the dipole in the electric field. The potential energy is a measure of the energy stored in the dipole due to its position in the electric field. This energy changes depending on the angle θ; when θ is zero (dipole aligned with the field), the potential energy is minimized, while at θ = 180° (dipole against the field), it is maximized. The negative sign in the equation reflects the nature of stable equilibrium when the potential energy is at its lowest.

Examples & Analogies

Consider pushing a shopping cart uphill. When you push from behind, you can imagine it like moving a dipole against the force (electric field). The higher you push it, the more potential energy you store in the cart, similar to how the dipole gains energy when oriented against the field.

Relevance of Potential Energy in Physics

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In understanding physical systems, this express potential energy plays a pivotal role in applications like molecular interactions, the stability of different molecular configurations, and the behavior of materials in electric fields.

Detailed Explanation

The potential energy of a dipole in an external field is not just a concept of theoretical physics; it has practical implications in various physical systems. Changes in this energy can influence how molecules behave in a field, especially in chemical reactions, as well as in designing materials for electronics that rely on dipole alignment and field interactions. The ability to remove or orient dipoles effectively is key in many technologies.

Examples & Analogies

Think about how magnets behave around each other. The stability of a magnet's orientation within a magnetic field is analogous to how dipoles behave in an electric field. Understanding this helps in designing better electronic devices, like sensors that use dipoles for detection.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Dipole Torque: Torque experienced by a dipole in a uniform electric field is expressed as t = p × E.

  • Work vs. Potential Energy: Work done against the electric field contributes to potential energy changes in the dipole.

  • Reference Points: Potential energy can vary based on the selected reference angle, typically set at θ0 = π/2.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • Changing the direction of an electric field results in energy changes in the alignment of dipoles.

  • In a real molecule, energy is released as heat when dipoles align with the applied electric field.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • Dipoles align, that's their action, work done leads to energy reaction.

📖 Fascinating Stories

  • Imagine a dipole in a dance, spinning with torque, seeking a chance to find its place with ease in the field, releasing energy when it finally yields.

🧠 Other Memory Gems

  • D for Dipole, T for Torque, W for Work, remember these for energy's quirk.

🎯 Super Acronyms

P.E.W. - Potential Energy from Work done against the electric field.

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Dipole

    Definition:

    A pair of equal and opposite electric charges separated by a distance.

  • Term: Torque

    Definition:

    A measure of the force that can cause an object to rotate about an axis.

  • Term: Potential Energy

    Definition:

    The energy stored in an object due to its position in a force field, with respect to a reference point.

  • Term: Electric Field

    Definition:

    A field around charged particles that exerts a force on other charged particles.