20.1 Magnetic Fields, Field Lines, and Force

Magnetic Fields and Properties

Magnetic fields are the invisible influence that magnets and moving charges create in the space around them. Understanding how these fields behave, how to visualize them, and how they exert forces on charges and currents is central to the rest of electromagnetism.

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Properties and Creation of Magnets

Every magnet has two poles: north and south. Opposite poles attract; like poles repel. No one has ever isolated a single magnetic pole (a magnetic monopole). If you snap a bar magnet in half, you get two smaller magnets, each with its own north and south pole.

Magnetic fields originate from moving electric charges. At the atomic level, this means current loops and electron spin both contribute. A magnetic dipole is the simplest source of a magnetic field, and you can think of a small current loop as one.

Ferromagnetic materials (iron, nickel, cobalt) contain tiny regions called magnetic domains, where atomic magnetic dipoles are already aligned. In an unmagnetized piece of iron, these domains point in random directions and cancel out. When you expose the material to a strong external field, the domains align, and the material becomes a permanent magnet.

Two weaker forms of magnetism also exist:

• Paramagnetic materials (aluminum, platinum) are weakly attracted to an external field because their atomic dipoles partially align with it.
• Diamagnetic materials (copper, water) are weakly repelled. The external field induces tiny opposing dipoles in the material.

The magnetization of a material is its magnetic moment per unit volume, and it describes how strongly the material responds to an applied field.

Magnetic Field Lines and Forces

Magnetic Field Lines

Magnetic field lines are a visual tool for mapping the direction and strength of a magnetic field.

• Lines point from north to south outside the magnet and from south to north inside it, forming continuous closed loops. They never start or stop (because monopoles don't exist) and they never cross.
• The density of field lines indicates field strength. Where lines are packed closely together (near the poles of a bar magnet, for example), the field is stronger.
• Around a current-carrying wire, field lines form concentric circles centered on the wire. Their direction follows the right-hand rule: point your thumb in the direction of conventional current, and your fingers curl in the direction the field lines wrap.

Magnetic flux is the total number of field lines passing through a surface, and it becomes important later when you study electromagnetic induction.

Force on a Moving Charge

A charged particle moving through a magnetic field experiences a force given by:

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

where $q$ is the charge, $\vec{v}$ is the velocity, and $\vec{B}$ is the magnetic field. This is the magnetic part of the Lorentz force.

Key features of this force:

• It is perpendicular to both $\vec{v}$ and $\vec{B}$. That means it can change the direction of a particle but never its speed (and therefore never its kinetic energy).
• If the charge is stationary ($v = 0$) or moving parallel to $\vec{B}$, the magnetic force is zero.
• The magnitude is $F = qvB\sin\theta$, where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$.

Right-hand rule for charges: Point your fingers in the direction of $\vec{v}$, curl them toward $\vec{B}$, and your thumb points in the direction of $\vec{F}$ for a positive charge. For a negative charge, the force points the opposite way.

Force on a Current-Carrying Wire

A straight wire carrying current $I$ in a magnetic field feels a force:

$\vec{F} = I\vec{L} \times \vec{B}$

where $\vec{L}$ is a vector along the wire (pointing in the direction of conventional current) with magnitude equal to the wire's length in the field.

The right-hand rule works the same way: point fingers along $I$, curl toward $\vec{B}$, and your thumb gives the force direction. The magnitude is $F = ILB\sin\theta$.

Circular Motion of Charged Particles

Because the magnetic force is always perpendicular to a particle's velocity, a charge moving in a uniform magnetic field follows a circular path. Setting the magnetic force equal to the centripetal force:

$qvB = \frac{mv^2}{r}$

Solving for the radius of the circular orbit:

$r = \frac{mv}{qB}$

The period of one full orbit is:

$T = \frac{2\pi m}{qB}$

Notice that $T$ does not depend on the particle's speed. A faster particle traces a larger circle but completes it in the same time. This is called cyclotron motion, and it's the principle behind cyclotron particle accelerators, where particles are repeatedly accelerated across a gap while the constant period keeps them in sync with an oscillating electric field.

Magnetic Moments and Torque

A current loop placed in a magnetic field doesn't just feel a net force (which may be zero in a uniform field); it feels a torque that tends to rotate it.

The magnetic moment of a current loop is:

$\vec{\mu} = IA\hat{n}$

where $I$ is the current, $A$ is the area enclosed by the loop, and $\hat{n}$ is the unit vector perpendicular to the plane of the loop (its direction is found by curling your right-hand fingers in the direction of current flow; your thumb gives $\hat{n}$).

The torque on the loop is:

$\vec{\tau} = \vec{\mu} \times \vec{B}$

The torque is maximum when the loop's plane is parallel to the field ($\vec{\mu}$ perpendicular to $\vec{B}$) and zero when $\vec{\mu}$ is aligned with $\vec{B}$. This is the operating principle behind electric motors: a current loop in a magnetic field experiences a torque that makes it spin.