Magnetic Field Effects: Complete Guide to Forces, Current Loops & Dipoles | HRK Physics Notes

Complete Guide to Magnetic Forces, Current Carrying Conductors & Magnetic Dipoles - B.Sc. Physics Edition 2015-16

Magnetic Field Lorentz Force Current Carrying Conductor Magnetic Dipole Reading Time: 25 min

📋 Table of Contents

• 1. Introduction to Magnetic Field
• 2. Magnetic Force on a Charged Particle
• 3. Magnetic Force on a Current Carrying Conductor
• 4. Torque on a Current Loop in Magnetic Field
• 5. The Magnetic Dipole
• 6. Potential Energy of Magnetic Dipole
• 7. Units of Magnetic Dipole Moment
• Frequently Asked Questions

Introduction to Magnetic Field

🧲 What is a Magnetic Field?

A magnetic field is the region or space around any magnet or moving charge within which its magnetic influence can be felt by other magnetic substances or moving charges.

🔬 Visualizing Magnetic Fields

• Magnetic field lines: Imaginary lines that represent the direction and strength of a magnetic field
• Field direction: From North pole to South pole outside a magnet
• Field strength: Indicated by the density of field lines
• Sources: Permanent magnets, moving charges, and current-carrying conductors

📏 Important Properties of Magnetic Fields

Magnetic field lines are continuous, closed loops that never intersect. They emerge from the North pole and enter the South pole of a magnet. The tangent to a field line at any point gives the direction of the magnetic field at that point.

Magnetic Force on a Charged Particle

⚡ Lorentz Force Law

When a charged particle moves in a magnetic field, it experiences a force perpendicular to both its velocity and the magnetic field direction. This is known as the magnetic component of the Lorentz force.

Magnetic Force Equation

For a charged object with charge \( q \) moving with velocity \( \vec{v} \) in a magnetic field \( \vec{B} \):

\[ \vec{F} = q(\vec{v} \times \vec{B}) \]

Magnitude of Magnetic Force

The magnitude of the magnetic force is given by:

\[ F = qvB \sin \theta \]

where \( \theta \) is the angle between the velocity vector and the magnetic field vector.

Maximum Magnetic Force

The maximum magnetic force occurs when the charged particle moves perpendicular to the magnetic field (\( \theta = 90^\circ \)):

\[ F_{max} = qvB \]

Definition of Magnetic Field Induction

From the maximum force equation, we can define magnetic field induction \( B \):

\[ B = \frac{F}{qv} \]

"The force on a unit positive charge moving perpendicular to the magnetic field with uniform velocity."

📐 Units of Magnetic Field

The SI unit of magnetic field induction \( \mathbf{B} \) is tesla (T), while in the cgs system it is measured in gauss (G).

\[ 1 \, \text{tesla} = 10^4 \, \text{gauss} \]

💡 Direction of Magnetic Force

The direction of the magnetic force is given by the right-hand rule:

• Point fingers in the direction of velocity \( \vec{v} \)
• Curl fingers toward magnetic field \( \vec{B} \)
• Thumb points in the direction of force \( \vec{F} \) for positive charges
• For negative charges, the force direction is opposite

Magnetic Force on a Current Carrying Conductor

🔌 Force on Current-Carrying Wire

When a current-carrying conductor is placed in a magnetic field, it experiences a force due to the magnetic forces acting on the moving charges within the conductor.

Setup and Parameters

Consider a straight wire of length \( L \) and cross-sectional area \( A \), carrying current \( I \) in a uniform magnetic field \( \vec{B} \).

Number of Charge Carriers

If \( n \) is the number of free charges per unit volume, each with charge \( e \), then the total number of charges in the wire is:

\[ N = nAL \]

Total Charge in the Wire

The total charge flowing through the wire is:

\[ q = nALe \]

Drift Velocity Calculation

If charges move with drift velocity \( v_d \) and cover length \( L \) in time \( t \):

\[ t = \frac{L}{v_d} \]

\[ I = \frac{q}{t} = \frac{nALe}{L/v_d} = nAev_d \]

\[ v_d = \frac{I}{nAe} \]

Force on Individual Charge

The force on one charge moving perpendicular to the magnetic field is:

\[ F' = ev_d B_\perp \]

Total Force on All Charges

The total force on all \( nAL \) charges is:

\[ F = nAL \cdot F' = nAL \cdot ev_d B_\perp \]

Substitute Drift Velocity

Substituting the expression for \( v_d \):

\[ F = nALe \left( \frac{I}{nAe} \right) B_\perp \]

\[ F = ILB_\perp \]

Vector Form

In vector form, the magnetic force on a current-carrying conductor is:

\[ \vec{F} = I(\vec{L} \times \vec{B}) \]

📐 General Case: Non-straight Wires

For wires that are not straight or in non-uniform magnetic fields, we divide the wire into infinitesimal elements of length \( d\vec{s} \):

\[ d\vec{F} = I(d\vec{s} \times \vec{B}) \]

The total force on the wire is obtained by integration:

\[ \vec{F} = \int d\vec{F} = \int I(d\vec{s} \times \vec{B}) \]

Torque on a Current Loop in Magnetic Field

🔄 Torque on Current Loop

When a current-carrying loop is placed in a magnetic field, magnetic forces on different segments of the loop can create a torque that tends to rotate the loop.

Rectangular Loop Setup

Consider a rectangular loop with sides of length \( a \) and \( b \), carrying current \( I \), suspended in a uniform magnetic field \( \vec{B} \).

Forces on Parallel Sides

Sides 1 and 3 (length \( b \)) are parallel to the magnetic field, so:

\[ \vec{L} \times \vec{B} = 0 \]

\[ F_1 = F_3 = 0 \]

Forces on Perpendicular Sides

Sides 2 and 4 (length \( a \)) are perpendicular to the magnetic field, so:

\[ F_2 = F_4 = F = IaB \sin 90^\circ = IaB \]

Direction of Forces

The forces \( F_2 \) and \( F_4 \) are equal in magnitude but opposite in direction, forming a couple that tends to rotate the loop.

Torque Calculation

The magnitude of torque is:

\[ \tau = (\text{Force}) \times (\text{Perpendicular Distance}) \]

\[ \tau = (F)(b \sin \theta) \]

\[ \tau = (IaB)(b \sin \theta) \]

\[ \tau = IabB \sin \theta \]

Area of the Loop

Since \( A = ab \) is the area of the loop:

\[ \tau = IAB \sin \theta \]

Loop with N Turns

For a coil with \( N \) turns, the torque becomes:

\[ \tau = NIAB \sin \theta \]

Vector Form

In vector form, the torque on a current-carrying loop is:

\[ \vec{\tau} = NI(\vec{A} \times \vec{B}) \]

📐 Torque Extremes

The torque has its maximum value when the field is perpendicular to the normal of the loop plane (\( \theta = 90^\circ \)):

\[ \tau_{max} = NIAB \]

The torque is zero when the field is parallel to the normal of the loop plane (\( \theta = 0^\circ \)).

The Magnetic Dipole

🧲 Magnetic Dipole Concept

A current-carrying loop behaves as a magnetic dipole - it has a North pole and a South pole separated by a small distance, creating a magnetic dipole moment.

Analogy with Electric Dipole

An electric dipole in an electric field experiences a torque:

\[ \vec{\tau} = \vec{p} \times \vec{E} \]

where \( \vec{p} = q\vec{d} \) is the electric dipole moment.

Magnitude of Electric Dipole Torque

The magnitude of torque on an electric dipole is:

\[ \tau = pE \sin \theta \]

Torque on Magnetic Dipole

Similarly, the torque on a magnetic dipole (current loop) in a magnetic field is:

\[ \tau = NIAB \sin \theta \]

Defining Magnetic Dipole Moment

By analogy with the electric case, we define the magnetic dipole moment \( \vec{\mu} \) as:

\[ \mu = NIA \]

Torque in Terms of Magnetic Moment

Using this definition, the torque becomes:

\[ \tau = \mu B \sin \theta \]

Vector Form

In vector form, the torque on a magnetic dipole is:

\[ \vec{\tau} = \vec{\mu} \times \vec{B} \]

📐 Magnetic Dipole Moment

The magnetic dipole moment \( \vec{\mu} \) is a vector associated with a magnet or current loop, defined as:

\[ \vec{\mu} = NI\vec{A} \]

where \( N \) is the number of turns, \( I \) is the current, and \( \vec{A} \) is the area vector (perpendicular to the plane of the loop).

Potential Energy of Magnetic Dipole

⚡ Energy of Magnetic Dipole

The work done to change the orientation of a magnetic dipole in a magnetic field is stored as potential energy of the magnetic dipole.

Work Done Against Torque

The work done to rotate the dipole from angle \( \theta_1 \) to \( \theta_2 \) is:

\[ W = \int_{\theta_1}^{\theta_2} \tau \, d\theta \]

Substitute Torque Expression

Using \( \tau = \mu B \sin \theta \):

\[ W = \int_{\theta_1}^{\theta_2} \mu B \sin \theta \, d\theta \]

Evaluate the Integral

\[ W = \mu B \int_{\theta_1}^{\theta_2} \sin \theta \, d\theta \]

\[ W = \mu B [-\cos \theta]_{\theta_1}^{\theta_2} \]

\[ W = \mu B (\cos \theta_1 - \cos \theta_2) \]

Potential Energy Definition

The potential energy is defined as the work done to rotate the dipole from the reference position (\( \theta_0 = 90^\circ \)) to angle \( \theta \):

\[ U = \int_{90^\circ}^{\theta} \tau \, d\theta \]

Calculate Potential Energy

\[ U = \mu B \int_{90^\circ}^{\theta} \sin \theta \, d\theta \]

\[ U = \mu B [-\cos \theta]_{90^\circ}^{\theta} \]

\[ U = \mu B (-\cos \theta + \cos 90^\circ) \]

\[ U = -\mu B \cos \theta \]

Vector Form

In vector form, the potential energy of a magnetic dipole is:

\[ U = -\vec{\mu} \cdot \vec{B} \]

📐 Potential Energy Extremes

The potential energy is minimum when the dipole is aligned with the field (\( \theta = 0^\circ \)):

\[ U_{min} = -\mu B \]

The potential energy is maximum when the dipole is anti-aligned with the field (\( \theta = 180^\circ \)):

\[ U_{max} = \mu B \]

Units of Magnetic Dipole Moment

📏 Determining the Units

The units of magnetic dipole moment can be derived from either the torque equation or the definition of magnetic moment.

From Potential Energy Equation

From the potential energy equation \( U = -\vec{\mu} \cdot \vec{B} \):

\[ \mu = \frac{U}{B} \]

Therefore, the unit of magnetic dipole moment is:

\[ \text{Unit of } \mu = \frac{\text{Energy unit}}{\text{Magnetic field unit}} \]

SI Units

In SI units:

\[ [\mu] = \frac{\text{Joule}}{\text{Tesla}} = \text{J/T} \]

From Magnetic Moment Definition

From the definition \( \mu = NIA \):

\[ [\mu] = \text{turn} \cdot \text{Ampere} \cdot \text{meter}^2 = \text{A·m}^2 \]

Equivalence of Units

These units are equivalent:

\[ 1 \, \text{J/T} = 1 \, \text{A·m}^2 \]

📐 Alternative Unit

Another common unit for magnetic dipole moment is the Bohr magneton, used in atomic physics:

\[ \mu_B = \frac{e\hbar}{2m_e} \approx 9.274 \times 10^{-24} \, \text{J/T} \]

Frequently Asked Questions

❓ Why does a charged particle moving parallel to a magnetic field experience no force?

When a charged particle moves parallel to a magnetic field, the angle between its velocity vector and the magnetic field vector is 0° or 180°. Since \( F = qvB \sin \theta \) and \( \sin 0^\circ = \sin 180^\circ = 0 \), the magnetic force is zero. The force is maximum when the particle moves perpendicular to the field (\( \theta = 90^\circ \)).

❓ What is the difference between magnetic field strength B and magnetic field intensity H?

Magnetic field strength B (magnetic flux density) is the actual magnetic field in a material, measured in tesla. Magnetic field intensity H is an auxiliary field that accounts for how the magnetic field is generated by free currents, measured in amperes per meter. They are related by \( \vec{B} = \mu_0(\vec{H} + \vec{M}) \), where \( \vec{M} \) is the magnetization of the material.

❓ Why does a current loop behave as a magnetic dipole?

A current loop behaves as a magnetic dipole because it produces a magnetic field pattern identical to that of a bar magnet. It has a North pole on one face and a South pole on the opposite face. When placed in an external magnetic field, it experiences a torque that tends to align its magnetic moment with the field, just like a compass needle.

❓ How is the direction of the magnetic dipole moment determined?

The direction of the magnetic dipole moment \( \vec{\mu} \) is given by the right-hand rule: if you curl the fingers of your right hand in the direction of current flow in the loop, your thumb points in the direction of \( \vec{\mu} \). Alternatively, \( \vec{\mu} \) is perpendicular to the plane of the loop.

❓ What practical applications use the torque on current loops in magnetic fields?

The torque on current loops in magnetic fields is the fundamental principle behind many devices including electric motors, galvanometers (for measuring current), and loudspeakers. In electric motors, the torque causes rotation that can be used to perform mechanical work.

📚 Continue Your Physics Journey

Understanding magnetic field effects is crucial for mastering electromagnetism and its applications. These comprehensive notes provide a solid foundation for further studies in electromagnetic induction, AC circuits, and modern physics.

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© House of Physics | Chapter 34: Magnetic Field Effects

These comprehensive notes are designed to help B.Sc. Physics students understand fundamental concepts of Magnetic Field Effects based on Halliday, Resnick and Krane

Author: Muhammad Ali Malik | Contact: +923016775811 | Email: aliphy2008@gmail.com