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4.9 - TORQUE ON CURRENT LOOP, MAGNETIC DIPOLE

Interactive Audio Lesson

Listen to a student-teacher conversation explaining the topic in a relatable way.

Introduction to Torque on Current Loop

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Teacher
Teacher

Good morning class! Today, we will learn about the torque experienced by a rectangular current loop in a magnetic field. Can anyone tell me what happens when a current-carrying wire is placed in a magnetic field?

Student 1
Student 1

I think it experiences a force.

Teacher
Teacher

Exactly! The force experienced depends on the current and the strength of the magnetic field. However, it's interesting to note that when placed in a uniform magnetic field, a loop experiences torque instead of a net force. Can someone explain why that's the case?

Student 2
Student 2

Is it because the forces on the opposite sides of the loop cancel each other out?

Teacher
Teacher

Right again! The forces on arms that are parallel to the field balance out, while forces on perpendicular arms produce a torque that causes rotation. Let's mark this key point: a loop in a uniform magnetic field rotates due to torque.

Student 3
Student 3

How do we calculate that torque?

Teacher
Teacher

Great question! Torque is given by the formula τ = IABsin(θ), where I is the current, A is the area, B is the magnetic field strength, and θ is the angle between the magnetic moment and the field direction.

Student 4
Student 4

What’s the magnetic moment, and why is it important?

Teacher
Teacher

The magnetic moment, m = IA, defines the strength and direction of the magnetic effect of the loop. It's crucial because it determines how the loop will interact with the magnetic field.

Teacher
Teacher

To summarize, we’ve established that a rectangular loop experiences torque when placed in a magnetic field due to the forces on its arms and that we calculate this torque using τ = IABsin(θ). Let’s move on to how this plays out in real applications.

Stable and Unstable Equilibrium

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Teacher
Teacher

Continuing from where we left off, let’s talk about the equilibriums when the loop aligns with the magnetic field. Can anyone guess what happens when the loop's magnetic moment is parallel to the magnetic field?

Student 1
Student 1

It should be in stable equilibrium, right?

Teacher
Teacher

Correct! In this position, any slight perturbation will cause a torque that restores it back. Now, what about when it's positioned antiparallel?

Student 2
Student 2

That would be unstable, since any small movement would increase the torque away from that position.

Teacher
Teacher

Exactly! In this unstable equilibrium, the loop will tend to rotate away from that position toward a more stable position. Here’s a mnemonic to remember: 'PULL for Parallel, FALL for Antiparallel.' Let’s test this knowledge further.

Student 3
Student 3

What practical applications come into play regarding these concepts?

Teacher
Teacher

Great segue! Understanding equilibrium is essential in devices like galvanometers and electric motors where controlling torque and rotation is paramount. To summarize, the loop's orientation determines if it’s in stable or unstable equilibrium based on its torque relative to the magnetic field.

Magnetic Moment and Its Significance

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Teacher
Teacher

I hope everyone is following along! Let's discuss the magnetic moment more closely. Can anyone tell me what we might use to define a loop’s magnetic moment?

Student 4
Student 4

It's the product of the current and the area of the loop, right?

Teacher
Teacher

Spot on! The magnetic moment is crucial because it expresses the strength and orientation of the loop in a magnetic field. Here’s a helpful picture: think of it as the 'arrow' of the loop in the magnetic field.

Student 1
Student 1

How does it relate to torque?

Teacher
Teacher

Great inquiry! The torque τ can be represented as τ = m × B, where m is the magnetic moment. This vector relationship shows us that torque depends on both the magnitude and orientation of m relative to B.

Student 2
Student 2

So does that mean magnetic moment can vary based on the angle?

Teacher
Teacher

Yes! The effectiveness of the magnetic moment in generating torque changes with the angle relative to the magnetic field. Finally, remember this: the stronger the magnetic moment, the more torque the loop experiences.

Teacher
Teacher

In conclusion, today we covered how torque is calculated, its connection to magnetic moments, and how these concepts are implemented in various electromagnetic devices. Any additional questions?

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

Quick Overview

The section discusses how a rectangular current loop in a uniform magnetic field experiences a torque and how this phenomenon is analogous to the behavior of electric dipoles in electric fields.

Standard

Understanding torque on a current loop is essential in electromagnetism as it explains how magnetic fields affect current-carrying loops. A rectangular loop experiences torque based on the angle with the magnetic field, and the magnetic moment plays a key role in determining this torque. The implications of this behavior are significant in many applications, including motors and galvanometers.

Detailed

Torque on a Current Loop

The rectangular loop carrying a steady current I experiences torque when placed in a uniform magnetic field B. The section introduces the fundamental concepts governing this behavior and highlights its similarity to electric dipoles in an electric field.

Key Points

  1. Torque Generation: When the rectangular loop is positioned such that the magnetic field is in its plane, it does not experience a net force but generates a torque. The torque is introduced through the forces on its arms, leading them to rotate in response to the magnetic field.
  2. Mathematical Formulation: The torque τ on the loop due to the forces can be expressed as:
    $$ au = I A B \sin(θ)$$
    where τ is the torque, I represents the current, A is the area of the loop, B is the magnetic field strength, and θ is the angle between the magnetic moment (m) and the magnetic field (B).
  3. Magnetic Moment: The magnetic moment of a current loop is defined as
    $$m = I A$$
    and points in a direction determined by the right-hand rule.
  4. Stable and Unstable Equilibrium: When the coil aligns parallel to the magnetic field, it is in a stable equilibrium; when in an antiparallel position, it is in an unstable equilibrium.
  5. Practical Implications: This behavior is crucial in many applications, including the design and operation of electric motors and galvanometers, where control over torque is necessary for functionality.

This section forms an essential part of understanding electromagnetic devices and their operation.

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Audio Book

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Torque on a Rectangular Current Loop

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We now show that a rectangular loop carrying a steady current I and placed in a uniform magnetic field experiences a torque. It does not experience a net force. This behaviour is analogous to that of electric dipole in a uniform electric field (Section 1.11).

We first consider the simple case when the rectangular loop is placed such that the uniform magnetic field B is in the plane of the loop. This is illustrated in Fig. 4.18(a). The field exerts no force on the two arms AD and BC of the loop. It is perpendicular to the arm AB of the loop and exerts a force F on it which is directed into the plane of the loop. Its magnitude is,

F = I b B

Similarly, it exerts a force F on the arm CD and F is directed out of the plane of the paper.

Thus, the net force on the loop is zero. There is a torque on the loop due to the pair of forces F and F.

Detailed Explanation

In this chunk, we learn that a rectangular current loop in a magnetic field doesn't experience net linear force but does experience torque. This occurs because the forces on different sides of the loop act oppositely, resulting in no net force but creating a tendency for the loop to rotate. The magnetic field interacts with the current, producing a force based on the geometry of the current loop.

Examples & Analogies

You can think of this like holding a pancake flat on a table. If you push down on one side and pull up on the opposite side at the same time, the pancake will not move laterally, but it will start to rotate. Similarly, the forces on the loop lead to rotation, or torque, rather than translation.

Torque Magnitude and Dependence on Angle

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Next, we consider the case when the plane of the loop is not along the magnetic field, but makes an angle with it. We take the angle between the field and the normal to the coil to be angle θ. The forces on the arms BC and DA are equal, opposite, and act along the axis of the coil, which connects the centres of mass of BC and DA. Being collinear along the axis, they cancel each other, resulting in no net force or torque. The forces on arms AB and CD are F and F. However, they are not collinear! This results in a couple as before. The torque is, however, less than the earlier case when the plane of the loop was along the magnetic field. This is because the perpendicular distance between the forces of the couple has decreased.

Detailed Explanation

Here, we understand how the angle between the plane of the loop and the magnetic field affects the torque. The torque is greatest when the loop is perpendicular to the magnetic field and decreases as the angle decreases. The concept of a couple is introduced, which is a pair of equal and opposite forces acting at a distance from one another, causing rotation.

Examples & Analogies

Imagine a door: when you push on the handle (which is far from the hinges), it swings wide open. If you push close to the hinges, it takes much less force to barely move. In this analogy, the door is like the current loop, and the force and distance from the 'hinge' (the pivot point of rotation) impact how effectively the door (loop) moves.

Defining Magnetic Moment and Torque Expression

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We define the magnetic moment of the current loop as, m = I A (4.22) where the direction of the area vector A is given by the right-hand thumb rule and is directed into the plane of the paper. Then as the angle between m and B is θ, Eqs. (4.20) and (4.21) can be expressed by one expression τ = m × B, where τ is the torque.

Detailed Explanation

In this chunk, we dive into defining the magnetic moment of the loop, which is a measure of its tendency to align with the magnetic field. The torque experienced by the loop can be expressed as the cross product of the magnetic moment and the magnetic field, indicating how these vectors interact.

Examples & Analogies

Think of a compass needle: it aligns itself with the Earth's magnetic field. The needle has a magnetic moment, and as it experiences torque, it turns to point north. Similarly, a current loop aligns itself when placed in a magnetic field, showing the relationship between torque and magnetic moment.

Equilibrium and Stability of the Loop

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From Eq. (4.23), we see that the torque τ vanishes when m is either parallel or antiparallel to the magnetic field B. This indicates a state of equilibrium as there is no torque on the coil. When m and B are parallel, the equilibrium is stable. Any small rotation of the coil produces a torque which brings it back to its original position.

Detailed Explanation

This segment discusses stability in terms of the orientation of the magnetic moment relative to the magnetic field. Understanding how the torque brings the loop back to its initial position illustrates the stability of configurations where the moment is aligned with the field, compared to those where it is opposite.

Examples & Analogies

Consider a pendulum: when it swings to the side (away from the vertical), gravity (analogous to torque) pulls it back to the lowest point (equilibrium). Just like a pendulum stabilizes at rest, a magnetic loop will tend to stabilize in a position where its magnetic moment aligns with the magnetic field.

Definitions & Key Concepts

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

Key Concepts

  • Torque: The rotational force experienced by a current-carrying loop in a magnetic field.

  • Magnetic Moment: Defines the strength and direction of the loop's magnetic effects, crucial for calculating torque.

  • Stable Equilibrium: The position where the loop remains stable when aligned with the magnetic field.

  • Unstable Equilibrium: The position where the loop tends to rotate away when misaligned with the magnetic field.

Examples & Real-Life Applications

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

Examples

  • A rectangular current loop in a uniform magnetic field produces torque, which can be calculated using τ = IABsin(θ).

  • In electric motors, the torque on current loops is harnessed to convert electrical energy into rotational mechanical energy.

Memory Aids

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

🎵 Rhymes Time

  • Torque on a loop brings rotation, in magnetic fields, it's a key foundation.

📖 Fascinating Stories

  • Imagine a fishing reel (the loop), spinning around in a river with a strong current (the magnetic field), the more you pull (increase current), the stronger the pull (torque) it feels on its path.

🧠 Other Memory Gems

  • PULL = Parallel Upward, FALL = Antiparallel Losing balance.

🎯 Super Acronyms

TOM = Torque on Magnetic moment.

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Torque

    Definition:

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

  • Term: Magnetic Moment

    Definition:

    A quantity that determines the torque exerted on a magnetic dipole in a magnetic field.

  • Term: Uniform Magnetic Field

    Definition:

    A magnetic field that has a constant strength and direction.

  • Term: Stable Equilibrium

    Definition:

    An equilibrium state where a small displacement returns the system to its original position.

  • Term: Unstable Equilibrium

    Definition:

    An equilibrium state where a small displacement causes the system to move away from its original position.