Magnetic Fields (Chapter 27): Study Notes

Magnetic Fields

Introduction to Magnetic Fields

Magnetic fields are fundamental to understanding the behavior of moving charges and currents in physics. They are vector fields that exert forces on moving charges and magnetic materials.

• Like poles of magnets repel each other, while unlike poles attract.

• The direction of the magnetic field at any point is the direction indicated by the north pole of a small compass needle placed at that point.

What Produces a Magnetic Field

Sources of Magnetic Fields

• Electromagnets: Created by passing a current through a wire.

• Permanent magnets: Materials that produce a persistent magnetic field.

• Both types attract small pieces of iron and, if suspended, align along the north-south direction.

These sources create a magnetic field B, which can exert a magnetic force FB on moving charges.

Moving Charge in a Magnetic Field

Conditions for Magnetic Force

• A charge must be moving to experience a magnetic force; stationary charges are unaffected.

• The velocity of the charge must have a component perpendicular to the magnetic field direction.

Definition and Formula for Magnetic Field

• The magnetic field B is defined by the force it exerts on a moving charge:

$F_B = |q| v B \sin \phi$

or in vector form:

$\vec{F}_B = q \vec{v} \times \vec{B}$

• Where $\phi$ is the angle between $\vec{v}$ and $\vec{B}$.

• SI unit of B: Tesla (T), where $1\ \text{T} = 1\ \text{N} \cdot \text{s}/\text{C} \cdot \text{m}$.

• Other unit: 1 gauss = $10^{-4}$ tesla.

Magnetic Field Lines

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

• The magnitude of B is proportional to the density of the field lines.

Motion of a Charged Particle in a Uniform Magnetic Field

• When a charged particle moves perpendicular to a uniform magnetic field, it undergoes circular motion due to the magnetic force acting as a centripetal force.

$F = qvB = m\frac{v^2}{r}$

Solving for the radius $r$:

$r = \frac{mv}{|q|B}$

The period $T$ of the circular motion is:

$T = \frac{2\pi m}{|q|B}$

The angular frequency $\omega$ is:

$\omega = \frac{|q|B}{m}$

• Note: The period does not depend on the speed or radius, only on $m$, $q$, and $B$.

Helical Paths

If the velocity of the particle forms an angle $\phi$ with the magnetic field, the motion is helical:

• Decompose velocity: $v_\parallel = v \cos \phi$, $v_\perp = v \sin \phi$

• The radius of the helix: $r = \frac{mv \sin \phi}{|q|B}$

• The pitch (distance advanced per turn): $p = T v_\parallel = \frac{2\pi m v \cos \phi}{|q|B}$

Applications: Mass Spectrometer

A mass spectrometer uses the motion of ions in a magnetic field to determine their mass:

$m = \frac{e r^2 B^2}{2V}$

• Where $e$ is the charge, $r$ is the radius of the path, $B$ is the magnetic field, and $V$ is the accelerating voltage.

Example Problems

• Example 1: Calculating the required magnetic field for a proton to just miss a plate:

• Given $v = 2.2 \times 10^6$ m/s, $r = 0.18$ m, $q = e$

• Solution: $B = \frac{mv}{qr} = 0.13$ T

• Example 2: Finding the angle and maximum $y$-position for a charged particle in a magnetic field.

• Solution: $\theta = 0^\circ$, $y_{max} = 0.29$ m

Magnetic Force on a Current-Carrying Wire

Force on a Straight Wire

• A current-carrying wire in a magnetic field experiences a force:

$F_B = i L B \sin \theta$

• Where $i$ is the current, $L$ is the length of the wire, $B$ is the magnetic field, and $\theta$ is the angle between $L$ and $B$.

Current Element in a Magnetic Field

• The force on a segment of wire (current element) is:

$F = I L B \sin \theta$

• Example: For a triangular loop in a uniform field, forces on sides can be calculated and net force determined.

Torque on a Current-Carrying Coil

Torque Formula

• A coil of $N$ turns, area $A$, carrying current $I$ in a magnetic field $B$ experiences a torque:

$\tau = N I A B \sin \phi$

• $NIA$ is called the magnetic moment.

• Example: For a rectangular loop with 75 turns, $I = 4.4$ A, $A = 0.7$ m$^2$, $B = 1.8$ T, $\phi = 35^\circ$:

• Torque $\tau = 170$ N·m; the angle will increase.

Application: Electric Motor

• Electric motors operate based on the torque exerted on current-carrying coils in magnetic fields.

Magnetic Materials

Types and Properties

• Magnetic materials can be unmagnetized or magnetized (aligned domains).

• Permanent magnets retain their magnetization after being exposed to a magnetic field.

Applications of Magnetic Fields

Examples

• Maglev trains: Use magnetic levitation for frictionless, high-speed travel.

• Magnetic recording: Magnetic fields are used to store information on tapes and disks.

Summary Table: Key Equations

Physical Quantity	Equation	Description
Magnetic Force on Charge	$F_B = |q| v B \sin \phi$	Force on charge $q$ moving with velocity $v$ in field $B$
Magnetic Force (vector)	$\vec{F}_B = q \vec{v} \times \vec{B}$	Vector form of magnetic force
Radius of Circular Path	$r = \frac{mv}{|q|B}$	Radius for charge in uniform $B$
Period of Circular Motion	$T = \frac{2\pi m}{|q|B}$	Time for one revolution
Force on Current-Carrying Wire	$F = i L B \sin \theta$	Force on wire of length $L$
Torque on Coil	$\tau = N I A B \sin \phi$	Torque on coil with $N$ turns
Mass Spectrometer	$m = \frac{e r^2 B^2}{2V}$	Mass of ion in spectrometer