Introduction to Magnetic Fields and Forces

Characteristics of Bar Magnets and Magnetic Dipoles

• Every bar magnet is defined as a magnetic dipole, meaning it always possesses two poles: a Magnetic North Pole and a Magnetic South Pole.
• A unique property of magnets is that they cannot be separated into individual monopoles. If a bar magnet is cut in half to create two smaller pieces, each of those new pieces will spontaneously have its own Magnetic North Pole and Magnetic South Pole.
• This process continues indefinitely; cutting those smaller pieces in half again results in new, even smaller magnets, each remaining a dipole with both a North and a South pole.
• Magnets are typically composed of ferromagnetic materials, such as iron, which allow for the alignment of internal magnetic moments.

Comparisons Between Magnetic and Electric Field Lines

• Electric field lines in an electric dipole are generated by separate point charges (positive and negative).
• Fundamental Rules for Electric Field Lines:
  • They originate on a positive charge or from infinity.
  • They terminate on a negative charge or at infinity.
  • Therefore, in an electric dipole, the lines always run from positive to negative.
• Magnetic field lines around a bar magnet appear very similar to electric dipole lines, but there are specific directional differences depending on the location relative to the magnet:
  • Outside/Around the magnet: Magnetic field lines run from the North Pole to the South Pole.
  • Through/Inside the magnet: Magnetic field lines run from the South Pole to the North Pole.
• The distinction of whether the question refers to the field "around" or "within" the magnet is critical for determining the correct direction.

Interactions Between Magnets and the Nature of Poles

• Magnetic interactions follow a similar logic to electric charges: like poles repel, and opposite poles attract.
  • North Pole to North Pole: Results in repulsion.
  • South Pole to South Pole: Results in repulsion.
  • North Pole to South Pole: Results in attraction (the magnets "stick").
  • This attraction occurs regardless of orientation (e.g., flipping both magnets so it is South to North still results in attraction).
• Repulsion and attraction also occur when magnets are aligned lengthwise; if like poles are adjacent (South to South and North to North), they repel.

Earth’s Magnetic Field and the Geographic/Magnetic Pole Distinction

• The naming of magnetic poles (North and South) is based on how they interact with the Earth's magnetic field.
• If a magnet is suspended by a string or floated on styrofoam in water, the North magnetic pole of the magnet will rotate to point toward the Geographic North Pole of the Earth.
• Because opposite poles attract, if the North-seeking pole of a bar magnet points toward the Earth's Geographic North, then the Earth's internal magnetic structure must have a Magnetic South Pole located at the Geographic North.
• Summary of Pole Locations:
  • Geographic North Pole: This is actually a Magnetic South Pole.
  • Geographic South Pole (Antarctica): This is actually a Magnetic North Pole.
• Geographic and Magnetic poles are not perfectly aligned:
  • The Magnetic South Pole is located in the northern Hudson Bay territory, not exactly at the Geographic North Pole.
  • The Magnetic North Pole is located off the coast of Antarctica, not in the center of the continent where the Geographic South Pole resides.
• These magnetic poles are not fixed; they move over time, and geological evidence suggests they reverse polarity every few million years.

The Origin of Magnetism and the Earth’s Core

• Magnetism in solid materials (like a bar magnet) is caused by the alignment of localized regions called magnetic moments.
• In ferromagnetic materials like iron, these moments can be aligned by proximity to another magnet, by scraping a magnet against the material, or by heating the substance to increase mobility and then cooling it to "freeze" the alignment.
• The Earth's core presents a puzzle: because scientists believe the core is molten liquid, it is impossible for magnetic moments to be permanently frozen in a single direction.
• Current scientific suspicion suggests that Earth's magnetic field is generated by electric currents running through the molten core, though the exact nature and cause of these currents remain unknown.

Review of Electric Field Forces

• To feel a force in an electric field, a particle must possess a charge.
• The magnitude of the electric force is calculated as:     $F = q \times E$
• Direction of electric force:
  • Positive charge: The force is in the same direction as the electric field.
  • Negative charge: The force is in the opposite direction of the electric field.

The Physics of Magnetic Forces on Moving Charges

• The symbol for a magnetic field is $\mathbf{B}$.
• Units of Magnetic Field:
  • The SI unit is the Tesla ($T$).
  • The Gauss ($G$) is a common non-SI unit because the Tesla is very large.
  • Conversion: $10^4\,G = 1\,T$.
  • The Earth’s magnetic field is approximately $0.5\,G$.
• Equation for Magnetic Force on a particle:     $F = q \times v \times B \times \sin(\theta)$
  • $q$: Charge of the particle.
  • $v$: Velocity of the particle.
  • $B$: Magnitude of the magnetic field.
  • $\theta$: The angle between the velocity vector ($\mathbf{v}$) and the magnetic field vector ($\mathbf{B}$).
• Necessary Conditions for Magnetic Force:
  1. The particle must be charged ($q \neq 0$).
  2. The particle must be in motion ($v \neq 0$). A stationary charge feels zero magnetic force.
  3. A component of the velocity must be perpendicular to the magnetic field. If velocity is parallel ($\theta = 0^{\circ}$) or anti-parallel ($\theta = 180^{\circ}$) to the field, then $\sin(\theta) = 0$ and the force is zero.
• Force is maximized when the particle moves perpendicularly to the magnetic field ($\theta = 90^{\circ}$), as $\sin(90^{\circ}) = 1$.

The Right-Hand Rule and Directional Conventions

• The magnetic force is the result of a cross product, meaning the force is orthogonal (perpendicular) to both the velocity vector and the magnetic field vector.
• Right-Hand Rule (RHR) Procedure (Chad's Convention):
  • Fingers: Point in the direction of the magnetic field ($\mathbf{B}$).
  • Thumb: Point in the direction of the velocity vector ($\mathbf{v}$).
  • Palm: The force ($\mathbf{F}$) comes out of the palm for a positive charge.
  • Back of Hand: The force ($\mathbf{F}$) comes out of the back of the hand for a negative charge (the opposite direction).
• Notation for 3D directions:
  • Out of the page: Represented by a dot ( $\odot$ ), like the tip of an arrow.
  • Into the page: Represented by an X ( $\otimes$ ), like the tail feathers of an arrow.
• Alternative RHR Methods (mentioned but not used):
  • The "Gun" Method: Index finger ($\mathbf{B}$), Thumb ($\mathbf{v}$), Middle finger ($\mathbf{F}$).
  • The "Curl" Method: Fingers point toward $\mathbf{v}$, curl toward $\mathbf{B}$, Thumb points toward $\mathbf{F}$.
• Left-Hand Rule Warning: While some use the left hand for negative charges, Chad advises against it to avoid confusion; always use the right hand and flip the direction for negative charges.

Numerical Example and Force Calculations

• Scenario: A $1.0\,\mu C$ positive charge enters a $2.0\,T$ magnetic field (directed into the page) with a velocity of $1200\,m/s$ (directed to the right).
• Step 1: Calculate the magnitude of the force ($q=1.0 \times 10^{-6}\,C$, $v=1200\,m/s$, $B=2.0\,T$, $\theta=90^{\circ}$).     $F = (1.0 \times 10^{-6}\,C) \times (1200\,m/s) \times (2.0\,T) \times \sin(90^{\circ})$$F = 2400 \times 10^{-6}\,N$$F = 2.4 \times 10^{-3}\,N$ (or $0.0024\,N$).
• Step 2: Determine initial direction using RHR.
  • Fingers point into the board ($\mathbf{B}$).
  • Thumb points to the right ($\mathbf{v}$).
  • Palm points up ($\mathbf{F}$).
  • Initial direction of force: Upward.

Circular Motion and Advanced Force Applications

• As the particle's trajectory changes due to the magnetic force, the direction of the velocity changes. Since the force must always be perpendicular to both the velocity and the field, the force direction also changes continuously.
• This results in a force that always points toward the center of a circular path, acting as a centripetal force.
• If the magnetic field is large enough, the particle will undergo Uniform Circular Motion.
• Extended Physics Applications:
  • Finding Acceleration ($a$): Use Newton’s Second Law: $F = m \times a$. Set $qvB\sin(\theta) = m \times a$.
  • Finding the Radius of the Circular Path ($r$): Use the centripetal force formula: $F_{c} = \frac{m \times v^2}{r}$.
  • Equation for radius: $qvB\sin(\theta) = \frac{m \times v^2}{r}$.