28.1 - Magnetic Fields and the Definition of $\vec B$

What Produces a Magnetic Field?

Because an electric field is produced by an electric charge, we might reasonably expect that a magnetic field is produced by a magnetic charge. Although individual magnetic charges (called magnetic monopoles) are predicted by certain theories, their existence has not been confirmed. How then are magnetic fields produced? There are two ways.

Moving electronically charged particles, such as a current in a wire, can be used to make an electromagnet. The current produces a magnetic field which can be controlled by controlling the flow of current through the wire.

The other way to produce a magnetic field is by means of elementary particles such as electrons because the particles have an intrinsic magnetic field around them. The magnetic fields of the electrons in certain materials add together to give a net magnetic field around the material. These materials are known as permanent magnets.

The Definition of $\vec B$

The magnetic field $\vec B$ is defined in terms of the magnetic force $\vec F_B$ exerted on a moving electrically charged test particle.

In principle, we do this by firing a charged particle through the point at which $\vec B$ is to be defined, using various directions and speeds for the particle and determining the force that acts on the particle at that point. After many such trials we would find that when the particle’s velocity $\vec v$ is along a particular axis through the point, force $\vec F_B$ is zero. For all other directions of $\vec v$, the magnitude of $\vec F_B$ is always proportional to $v \sin \phi$, where $\phi$ is the angle between the zero-force axis and the direction of $\vec v$.

We can then define a magnetic field $\vec B$ to be a vector quantity that is directed along the zero-force axis. We can next measure the magnitude of $\vec F_B$ when $\vec v$ is directed perpendicular to that axis and then define the magnitude of $\vec B$ in terms of that force magnitude:

where $q$ is the charge of the particle.

Finding the Magnetic Force on a Particle

The force $\vec F_B$ on a particle from a magnetic field can be represented as a cross product:

that is, the force $\vec F_B$ on the particle is equal to the charge $q$ times the cross product of its velocity $\vec v$ and the field $\vec B$.

Thus, we can write the magnitude of $\vec F_B$ as:

where $\phi$ is the angle between the directions of the velocity $\vec v$ and the magnetic field $\vec B$.

28.2 - Crossed Fields: Discovery of the Electron

If a charged particle moves through a region containing both an electric field and a magnetic field, it can be affected by both an electric force and a magnetic force.

If the fields are perpendicular to each other, they are said to be crossed fields.

If the forces are in opposite directions, a particular speed will result in no deflection of the particle.

28.3 - Crossed Fields: The Hall Effect

When a uniform magnetic field $B$ is applied to a conducting strip carrying current $i$, with the field perpendicular to the direction of the current, a Hall-effect potential difference $V$ is set up across the strip.

The electric force $\vec F_E$ on the charge carriers is then balanced by the magnetic force $\vec F_B$ on them.

The number density n of the charge carriers can then be determined from:

where $l$ is the thickness of the strip (parallel to $\vec B$).

When a conductor moves through a uniform magnetic field $\vec B$ at speed $v$, the Hall-effect potential difference $V$ across it is:

where $d$ is the width perpendicular to both velocity $\vec v$ and field $\vec B$.

28.4 - A Circulating Charged Particle

A charged particle with mass $m$ and charge magnitude $|q|$ moving with velocity $\vec v$ perpendicular to a uniform magnetic field $\vec B$ will travel in a circle.

Applying Newton’s second law to the circular motion yields:

from which we find the radius $r$ of the circle to be

The frequency of revolution $f$, the angular frequency $\omega$, and the period of the motion $T$ are given by:

If the velocity of the particle has a component parallel to the magnetic field, the particle moves in a helical path around the field vector $\vec B$.

28.5 - Cyclotrons and Synchrotrons

A cyclotron is a form of particle accelerator in which particles are made to circulate using an an electrical oscillator to induce magnetic fields that slowly increase the speed of the particles (e.g. protons).

Frequency

The key to the operation of the cyclotron is that the frequency $f$ at which the proton circulates in the magnetic field (and that does not depend on its speed) must be equal to the fixed frequency $f_{osc}$ of the electrical oscillator:

which can be used in combination with the above equation for frequency to obtain:

The Proton Synchrotron

At proton energies above 50 MeV, the conventional cyclotron begins to fail because one of the assumptions of its design—that the frequency of revolution of a charged particle circulating in a magnetic field is independent of the particle’s speed—is true only for speeds that are much less than the speed of light.

The proton synchrotron corrects for the relativity and feasibility problems of a traditional cyclotron. The magnetic field $B$ and the oscillator frequency $f_{osc}$, instead of having field values as in the conventional cyclotron, are made to vary with time during accelerating cycle.

28.6 - Magnetic Force on a Current-Carrying Wire

A straight wire carrying a current $i$ in a uniform magnetic field experiences a sideways force:

The force acting on a current element $i \space d\vec L$ in a magnetic field is:

The direction of the length vector $\vec L$ or $d\vec L$ is that of the current $i$.

28.7 - Torque on a Current Loop

Various magnetic forces act on the sections of a current-carrying coil lying in a uniform external magnetic field, but the net force is zero.

The net torque acting on the coil has a magnitude given by:

where $N$ is the number of turns in the coil, $A$ is the area of each turn, $i$ is the current, $B$ is the field magnitude, and $\theta$ is the angle between the magnetic field $\vec B$ and the normal vector to the coil $\vec n$.

28.8 - The Magnetic Dipole Moment

A coil (of area $A$ and $N$ turns, carrying current $i$) in a uniform magnetic field will experience a torque given by:

Here $\vec \mu$ is the magnetic dipole moment of the coil, with magnitude $\mu = NiA$ and direction given by the right-hand rule.

The orientation energy of a magnetic dipole in a magnetic field is given by:

If an external agent rotates a magnetic dipole from an initial orientation $\theta_i$ to some other orientation $\theta_f$ and the dipole is stationary both initially and finally, the work $W_a$ done on the dipole by the agent is: