Magnetic Fields, Forces, and Applications: Study Notes

Bar Magnets

Magnetic Poles and Interactions

Bar magnets possess two distinct poles: North (N) and South (S). The fundamental interactions between these poles are:

• Unlike poles attract: The north pole of one magnet attracts the south pole of another.

• Like poles repel: Two north poles or two south poles repel each other.

Magnetic poles always come in pairs. Unlike electric charges, you cannot isolate a single magnetic pole. Cutting a magnet in half always yields two smaller dipoles, each with a north and south pole.

The Earth's Magnetic Field

Structure and Properties

The Earth acts as a giant bar magnet, generating a magnetic field that surrounds the planet. Key points include:

• Geographic vs. Magnetic Poles: The geographic north pole is actually a magnetic south pole, which attracts the north pole of a compass.

• Magnetic Field Lines: These lines show the direction a compass would point at any location. They emerge from the magnetic north pole and enter the magnetic south pole.

• Origin: The Earth's magnetic field is thought to be generated by electric currents in its molten core.

Electromagnetism

Connection Between Electricity and Magnetism

Electric currents produce magnetic fields. This relationship is demonstrated by placing a compass near a current-carrying wire:

• When no current flows, the compass needle points north.

• When current flows, the compass needle deflects, indicating the presence of a magnetic field around the wire. The direction of deflection depends on the direction of the current.

Magnetic Fields

Field Vectors and Field Lines

A magnetic field is a vector field, denoted by B, defined at every point in space. Important properties:

• At each point, the field line is tangent to the magnetic field vector B.

• Compasses align with the direction of B.

• The strength of the field is greater where field lines are more densely packed.

• Field lines point away from north poles and toward south poles.

Visualizing Magnetic Fields

• Magnetic field vectors point in the direction of compass needles.

• Stronger fields are represented by longer vectors.

Field Lines Around Magnets and Currents

• Field lines leave the north pole and enter the south pole of a bar magnet.

• Between flat, parallel magnetic poles, the field is nearly uniform.

• For a straight current-carrying wire, field lines form concentric circles around the wire.

• For a current-carrying loop or solenoid, the field resembles that of a bar magnet.

Magnetic Field Superposition

Combining Magnetic Fields

The total magnetic field at any point is the vector sum of the fields from all magnetic sources nearby. This is analogous to the superposition principle for electric fields.

• Example: Two solenoids producing nearly uniform fields will have their fields add vectorially.

Magnetic Forces

Force on Moving Charges

Magnetic fields exert a force on charged particles moving through them. The magnitude of the force is given by:

• Proportional to the charge q

• Proportional to the speed v of the particle

• Proportional to the magnitude of the magnetic field B

• Depends on the angle φ between velocity and field

Formula:

$F = |q|vB \sin \phi$

The force acts perpendicular to both the velocity and the magnetic field.

Unit of B: Tesla (T), where $1\,\text{T} = 1\,\text{N}/(\text{A}\cdot\text{m})$

Right-Hand Rule

The direction of the magnetic force on a positive charge is given by the right-hand rule:

• Point fingers in the direction of velocity v.

• Rotate fingers toward the direction of B.

• Thumb points in the direction of the force F.

• If the charge is negative, the force direction is opposite.

Motion of Charged Particles in a Magnetic Field

Circular Motion

A charged particle moving perpendicular to a uniform magnetic field experiences a force perpendicular to its velocity, resulting in circular motion.

• Force magnitude: $F = qvB$

• Centripetal force: $F = ma_{\text{rad}}$

• Radius of circle: $R = \frac{mv}{qB}$

Work done by the magnetic field on the particle is zero, since the force is always perpendicular to the velocity.

Force on a Current-Carrying Conductor

Derivation and Formula

A wire carrying current in a magnetic field experiences a force due to the motion of charge carriers:

• Force on each charge: $f = qv_dB$ (where $v_d$ is drift velocity)

• Total charge in rod: $Q = \frac{I\ell}{v_d}$

• Total force: $F = Qv_dB = I\ell B$

• If the field is not perpendicular: $F = I\ell B \sin \phi$

Force and Torque on a Current Loop

Torque on a Rectangular Loop

A current loop in a uniform magnetic field experiences a torque that tends to align the loop with the field:

• Force on each side: $F = IaB \sin \phi$

• Total torque: $\tau = IAB \sin \phi$ (where $A$ is area of loop)

• Torque is maximal when $\phi = 90^\circ$ and zero when $\phi = 0^\circ$ or $180^\circ$.

Equilibrium Conditions

• Stable equilibrium: $\phi = 0^\circ$

• Unstable equilibrium: $\phi = 180^\circ$

Solenoid in a Uniform Field

Torque on a Solenoid

A solenoid in a uniform magnetic field experiences a torque that tends to align it with the field, similar to a bar magnet.

Magnetic Field from a Long Straight Wire

Field Generation and Formula

Current-carrying wires generate magnetic fields. For a long straight wire:

• Magnitude of field at distance r:

$B = \frac{\mu_0 I}{2\pi r}$

• Permeability of vacuum: $\mu_0 = 4\pi \times 10^{-7}\,\text{T}\cdot\text{m}/\text{A}$

Right-hand rule applies: thumb in direction of current, fingers curl in direction of field lines.

Magnetic Field from a Loop

Field Generation by Current Loops

• Single loop: $B = \frac{\mu_0 I}{2r}$

• N loops: $B = \frac{\mu_0 NI}{2r}$

Right-hand rule applies for direction of field.

Solenoid

Field Strength Inside a Solenoid

• Field inside: $B = \mu_0 n I$

• $n$ = turns per unit length

• Field is uniform inside, much stronger than outside.

Typical Field Strengths

Comparison Table

Source	Field Strength (B)
Wire (1 cm away, 10 A)	$10^{-4}$ T = 1 Gauss
Solenoid (200 turns, 10 cm, 10 A)	$10^{-3}$ T = 10 Gauss
Fridge magnet (close to pole)	$10^{-2}$ T = 100 Gauss
Earth's magnetic field	$5 \times 10^{-5}$ T = 0.5 Gauss
Electromagnet (iron core)	$1$T = $10^4$ Gauss
Superconducting electromagnet (LHC)	$10$T = $10^5$ Gauss

Wire in a Field

Behavior of a Metal Bar in a Magnetic Field

A metal bar connected to a battery and placed between the poles of a horseshoe magnet will experience a force when current flows, causing it to swing outward, away from the magnet.

Summary Table: Key Equations

Situation	Equation
Force on moving charge	$F = |q|vB \sin \phi$
Force on current-carrying wire	$F = I\ell B \sin \phi$
Torque on current loop	$\tau = IAB \sin \phi$
Field from straight wire	$B = \frac{\mu_0 I}{2\pi r}$
Field from loop	$B = \frac{\mu_0 I}{2r}$
Field inside solenoid	$B = \mu_0 n I$
Radius of particle path	$R = \frac{mv}{qB}$

Additional info: These notes cover topics from Chapter 20: Magnetic Fields and Magnetic Forces, and Chapter 21: Electromagnetic Induction, as well as applications relevant to college-level physics.