Understanding Angular Momentum, Magnetic Dipole Moments, Torque, and Their Interaction with Magnetic Fields
Solved Example Included for NEET, JEE, WBJEE
Introduction
In physics, understanding the relationship between angular momentum, magnetic dipole moments, torque, and external magnetic fields is crucial, especially when solving problems involving charged particles like electrons. This article explains these concepts step-by-step and demonstrates how to tackle questions where an electron revolves in a magnetic field.
What is Angular Momentum?
Angular momentum L⃗ is a measure of the rotational motion of an object. It’s analogous to linear momentum but in the context of circular or rotational motion. For a particle moving in a circular orbit, angular momentum is defined as:
L⃗= r⃗×p⃗,
where:
r⃗ is the position vector of the particle relative to the axis of rotation,
p⃗=m v⃗ is its linear momentum (mass times velocity),
The direction of L⃗ is given by the right-hand rule (explained below).
Angular momentum is conserved in the absence of external torques, making it a fundamental quantity in physics.
What is Angular Momentum -
What is Torque?
Torque (T⃗) measures the rotational force that causes an object to rotate about an axis. It is a vector quantity and is defined as:
T⃗=r⃗×F⃗,
where:
r⃗ is the position vector from the axis of rotation to the point where the force is applied,
F⃗ is the applied force,
The right-hand rule gives the direction of T⃗ .
Key Points About Torque:
Direction of Torque:
The torque vector points perpendicular to the plane formed by r⃗ and F⃗.
Effect of Torque:
Torque causes angular acceleration, changing the angular momentum of the object.
Magnitude of Torque:
The magnitude of T⃗ (Vector) is: T=r F sinθ where θ is the angle between r⃗ and F⃗.
In problems involving magnetic fields, torque often comes into play when magnetic dipoles interact with the external field.
What is the Right-Hand Rule?
The right-hand rule is a simple and intuitive way to determine the direction of a vector resulting from a cross product, such as angular momentum L⃗=r⃗×p⃗ or torque T⃗=μ⃗×B⃗ .
Steps for Using the Right-Hand Rule:
For Angular Momentum (L⃗):
Point your right-hand fingers in the direction of r⃗ (the position vector of the particle).
Curl your fingers toward p⃗ (the linear momentum vector or the velocity vector).
Your thumb will point in the direction of L⃗ (angular momentum).
For Torque (T⃗):
Point your right-hand fingers in the direction of μ⃗ (magnetic dipole moment).
Curl your fingers toward B⃗ (magnetic field vector).
Your thumb will point in the direction of T⃗ (torque).
What is a Magnetic Dipole Moment?
A magnetic dipole moment (μ⃗ ) is a measure of the strength and direction of a magnetic source, typically caused by a current loop or the motion of a charged particle. For a charged particle like an electron in circular motion:
μ⃗=I⋅Area⋅n^
where:
I is the current due to the motion of the charge,
A( Area ) is the area of the loop traced by the particle,
n^ is a unit vector perpendicular to the plane of the loop, determined by the right-hand rule for current.
For an electron, the current I is opposite to its motion because the electron carries a negative charge.
What is Magnetic Dipole Moment
Why is μ⃗ Antiparallel to L⃗ ?
The relationship between μ⃗ (magnetic dipole moment) and L⃗ (angular momentum) is:
μ⃗=−(e/2m) L⃗.
Here’s why:
The electron is negatively charged, so its current flows in the opposite direction to its motion.
The direction of μ⃗ is determined by the right-hand rule for current, while L⃗ is determined by the right-hand rule for angular momentum.
The negative charge reverses the direction of μ⃗ compared to L⃗.
As a result, μ⃗ is antiparallel to L⃗.
Torque on a Magnetic Dipole in a Magnetic Field
When a magnetic dipole (μ⃗) is placed in an external magnetic field (B⃗), it experiences a torque:
T⃗=μ⃗×B⃗.
Key Points:
Direction of Torque:
The torque T⃗ is perpendicular to both μ⃗ and B⃗, as given by the right-hand rule.
Effect of Torque:
The torque causes the dipole to rotate about the direction of B⃗.
Since μ⃗ is antiparallel to L⃗, the torque T⃗ is also perpendicular to the angular momentum L⃗.
Solving Problems: Step-by-Step Approach
Here’s how to approach questions involving angular momentum, magnetic dipole moments, and torque:
1. Understand the System:
Identify the particle’s motion and determine its angular momentum (L⃗).
Calculate the magnetic dipole moment (μ⃗=−e/2mL⃗).
2. Analyze the Magnetic Field:
Check the direction of the external magnetic field (B⃗).
Note whether B⃗ is parallel, perpendicular, or at an angle to the plane of the orbit.
3. Compute the Torque:
Use T⃗= μ⃗×B⃗ to find the torque.
Remember that the direction of T⃗ is determined by the right-hand rule.
4. Interpret the Result:
If asked about the angle between T⃗ and L⃗, remember that μ⃗ is antiparallel to L⃗, so T⃗ is perpendicular to L⃗.
Example Problem
Question: An electron revolves around the nucleus in a circular path with angular momentum L⃗. A uniform magnetic field B⃗ is applied perpendicular to the plane of its orbit. If the electron experiences a torque T⃗, determine the relationship between T⃗ and L⃗.
Solution:
The electron has an angular momentum L⃗ due to its circular motion.
Its magnetic dipole moment μ⃗ is antiparallel to L⃗ μ⃗=−(e/2m) L⃗
The torque on the electron is: T⃗=μ⃗×B⃗..
Since μ⃗ is antiparallel to L⃗, and B⃗ is perpendicular to the plane of the orbit, the torque T⃗ is perpendicular to both μ⃗ and B⃗.
Therefore, T⃗ is perpendicular to L⃗.
Answer: T⃗⊥L⃗.
Key Takeaways
Angular momentum L⃗ represents rotational motion and is conserved in the absence of external torques.
Torque (T⃗ is the rotational force that changes angular momentum, acting perpendicular to the applied force and position vector.
A charged particle in circular motion generates a magnetic dipole moment μ⃗, which is antiparallel to L⃗ for an electron.
The torque T⃗ experienced by a magnetic dipole in a magnetic field is given by T⃗= μ⃗×B⃗.
The right-hand rule is a crucial tool for determining the directions of L⃗, μ⃗, and T⃗.
By mastering these steps and using the right-hand rule, students can confidently solve problems involving angular momentum, torque, and magnetic fields.
