6.2 Magnetic force on moving charges

Magnetic fields

Magnetic fields describe the region of space where magnetic forces act on moving charges and magnetic materials. They're central to understanding how charged particles behave in everything from lab equipment to Earth's magnetosphere.

Sources of magnetic fields

• Permanent magnets generate static fields due to aligned magnetic domains within the material.
• Moving electric charges produce magnetic fields. This includes electric currents flowing through wires.
• Earth's magnetic field originates from convection currents in its liquid outer core (a process called the geodynamo).
• Electromagnets create controllable magnetic fields by passing current through coils of wire. Increasing the current or the number of coils strengthens the field.

Magnetic field lines

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

• They form closed loops, running from the north pole to the south pole outside a magnet and continuing through the interior. They never intersect.
• The density of field lines indicates field strength: closely packed lines mean a stronger field.
• The tangent to a field line at any point gives the direction of the field at that location.

Magnetic flux

Magnetic flux measures how much magnetic field passes through a given surface area. Think of it as "how many field lines thread through a loop."

$\Phi_B = BA \cos\theta$

Here, $B$ is the magnetic field strength, $A$ is the area of the surface, and $\theta$ is the angle between the field and the normal (perpendicular) to the surface. Units are webers (Wb), equivalent to T·m². Changes in magnetic flux are what drive electromagnetic induction (Faraday's law), which you'll encounter later.

Charged particles in motion

The way a charged particle moves through a magnetic field depends entirely on the relationship between its velocity and the field direction.

Velocity vs. magnetic field

• If the particle's velocity is parallel to the magnetic field, there is zero magnetic force. The particle just keeps moving in a straight line.
• If the velocity is perpendicular to the field, the magnetic force is at its maximum.
• At any angle in between, only the velocity component perpendicular to the field contributes to the force. The parallel component is unaffected.

Lorentz force equation

The Lorentz force equation gives the total electromagnetic force on a charged particle:

$\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})$

This combines the electric force $q\mathbf{E}$ (which acts whether the charge is moving or not) with the magnetic force $q\mathbf{v} \times \mathbf{B}$ (which only acts on moving charges). In situations where there's no electric field, the equation reduces to just the magnetic force term.

Force on moving charges

Magnitude of magnetic force

The magnitude of the magnetic force on a single moving charge is:

$F = qvB\sin\theta$

where:

• $q$ = charge of the particle (in coulombs)
• $v$ = speed of the particle (in m/s)
• $B$ = magnetic field strength (in tesla)
• $\theta$ = angle between the velocity vector and the magnetic field

When $\theta = 90°$, $\sin\theta = 1$ and the force is maximized. When $\theta = 0°$ (velocity parallel to field), the force is zero.

Direction of magnetic force

The magnetic force is always perpendicular to both the velocity and the magnetic field. This means it can change the particle's direction but never its speed.

Because the force is perpendicular to velocity, a charged particle in a uniform magnetic field will curve. Positive charges curve one way; negative charges curve the opposite way.

Right-hand rule

To find the direction of the magnetic force on a positive charge:

1. Point your fingers in the direction of the particle's velocity ($\mathbf{v}$).
2. Curl your fingers toward the magnetic field ($\mathbf{B}$).
3. Your thumb points in the direction of the force ($\mathbf{F}$).

For a negative charge, follow the same steps, then reverse the direction your thumb points. (Some students find it easier to use a "left-hand rule" for negative charges instead.)

Circular motion in magnetic fields

When a charged particle moves perpendicular to a uniform magnetic field, the magnetic force acts as a centripetal force, pulling the particle into a circular path.

Radius of circular path

Setting the magnetic force equal to the centripetal force:

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

Solving for $r$:

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

So the radius increases with greater mass or speed, and decreases with stronger fields or larger charges. For example, a proton and an electron entering the same magnetic field at the same speed will follow very different circles: the proton's radius will be much larger because its mass is about 1836 times the electron's mass.

Period of rotation

The time for one complete revolution is:

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

Notice something important: the period does not depend on the particle's speed or the radius of its path. A faster particle traces a bigger circle but covers the extra distance in the same time. This velocity-independence is what makes cyclotrons work.

Cyclotron frequency

The angular frequency of the circular motion is:

$\omega = \frac{qB}{m}$

This is called the cyclotron frequency. In a cyclotron particle accelerator, an alternating electric field is tuned to this frequency so that it accelerates the particle at exactly the right moments during each orbit.

Applications of magnetic force

Mass spectrometry

A mass spectrometer separates ions by their mass-to-charge ratio ($m/q$). Ions are first accelerated through a known potential difference, then enter a region of uniform magnetic field where they follow circular paths. From $r = mv/(qB)$, ions with different masses curve by different amounts, landing at different positions on a detector. This technique is used in chemical analysis, isotope identification, forensic science, and pharmaceutical research.

Hall effect

When current flows through a conductor and a magnetic field is applied perpendicular to that current, the magnetic force pushes charge carriers to one side of the conductor. This creates a measurable voltage difference across the conductor called the Hall voltage. Hall effect sensors are widely used to measure magnetic field strength, detect current, and sense position in automotive and industrial systems.

Particle accelerators

Particle accelerators use electric fields to speed up charged particles and magnetic fields to steer and focus the beams. Types include linear accelerators (linacs), cyclotrons, and synchrotrons. Beyond fundamental physics research (like at CERN), they're used in radiation therapy for cancer treatment and in materials science.

Magnetic force on current-carrying wires

A current-carrying wire is just a collection of moving charges, so it experiences a force in a magnetic field too.

Force on straight wires

$F = ILB\sin\theta$

where $I$ is the current, $L$ is the length of wire in the field, $B$ is the field strength, and $\theta$ is the angle between the current direction and the field. The direction of the force follows the right-hand rule, with your fingers pointing along the current direction instead of velocity. This principle is how electric motors and loudspeakers work.

Force between parallel wires

Two parallel wires carrying current exert forces on each other:

• Currents in the same direction → wires attract
• Currents in opposite directions → wires repel

The force per unit length between them is:

$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$

Here, $\mu_0 = 4\pi \times 10^{-7} \text{ T·m/A}$ is the permeability of free space, $I_1$ and $I_2$ are the currents, and $d$ is the distance between the wires. This relationship was historically used to define the ampere in the SI system.

Motion of charged particles

Helical motion

If a charged particle enters a magnetic field with velocity components both parallel and perpendicular to the field, it follows a helical (corkscrew) path. The perpendicular component produces circular motion, while the parallel component carries the particle forward along the field direction, unaffected by the magnetic force. The pitch of the helix (distance between loops) depends on how large the parallel velocity component is relative to the perpendicular one. This type of motion is seen in cosmic rays spiraling along Earth's magnetic field lines.

Drift velocity

In non-uniform or combined fields, charged particles experience slow, net displacement called drift. Common types include:

• E×B drift: occurs when both electric and magnetic fields are present; the particle drifts perpendicular to both fields.
• Gradient drift: occurs when the magnetic field strength varies across space.
• Curvature drift: occurs when field lines are curved.

These drifts are critical in plasma physics, particularly in fusion reactor design, where particle drifts can cause plasma to escape confinement.

Magnetic mirrors

A magnetic mirror is a region where the magnetic field strength increases. As a charged particle spirals into a stronger field region, the force gradually reverses the particle's parallel motion, reflecting it back. This is based on conservation of the particle's magnetic moment.

Magnetic mirrors occur naturally in Earth's magnetosphere (trapping charged particles in the Van Allen radiation belts) and are used in some plasma confinement devices. Particles with too much parallel velocity relative to their perpendicular velocity can escape through the mirror, creating what's called a loss cone.

Energy considerations

Work done by magnetic force

The magnetic force does no work on a moving charged particle. Since the force is always perpendicular to the velocity, the displacement at any instant is perpendicular to the force:

$W = \mathbf{F} \cdot \mathbf{d} = 0$

This means magnetic fields can redirect a particle's motion but cannot change its kinetic energy or speed. If a particle speeds up in a device like a cyclotron, it's the electric field doing the work, not the magnetic field.

Magnetic potential energy

For a single charged particle, magnetic potential energy isn't really defined in the usual sense. However, for a magnetic dipole (like a current loop or bar magnet) in an external field, the potential energy is:

$U = -\mathbf{m} \cdot \mathbf{B} = -mB\cos\theta$

where $\mathbf{m}$ is the magnetic dipole moment and $\theta$ is the angle between the dipole and the field. The energy is lowest (most stable) when the dipole is aligned with the field, and highest when it's anti-aligned. This explains why compass needles rotate to align with magnetic fields.

Magnetic force vs. electric force

Similarities and differences

Combined electromagnetic fields

Many real-world situations involve both electric and magnetic fields acting simultaneously. The full Lorentz force equation $\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})$ handles these cases. One important result is E×B drift, where a charged particle in crossed electric and magnetic fields drifts at a velocity of $v_d = E/B$, perpendicular to both fields. This drift is independent of the particle's charge and mass, which makes it useful for velocity selectors in devices like mass spectrometers.

Experimental demonstrations

Cathode ray tube

The cathode ray tube (CRT) fires a beam of electrons that can be deflected by magnetic fields. By observing how much the beam bends, you can extract information about the electrons. CRTs were the basis of older television and computer monitor technology, and they remain a useful teaching tool for visualizing how magnetic fields steer charged particles.

Thomson's e/m experiment

J.J. Thomson measured the electron's charge-to-mass ratio ($e/m$) by using crossed electric and magnetic fields:

1. An electric field deflected the electron beam in one direction.
2. A magnetic field was applied to deflect the beam in the opposite direction.
3. The fields were adjusted until the beam traveled straight, meaning the electric and magnetic forces balanced: $qE = qvB$, giving $v = E/B$.
4. With the velocity known, the magnetic field alone was used to bend the beam in a circle, and $e/m = v/(rB)$ was calculated from the measured radius.

This experiment demonstrated that cathode rays were streams of negatively charged particles (electrons) and was a landmark in the discovery of the electron.

Bainbridge mass spectrometer

The Bainbridge mass spectrometer combines a velocity selector with a magnetic deflection region:

1. Ions pass through crossed electric and magnetic fields. Only ions with velocity $v = E/B$ pass straight through (the forces balance for that specific speed).
2. These velocity-selected ions then enter a region of uniform magnetic field, where they follow semicircular paths.
3. Ions of different masses land at different positions on a detector, since $r = mv/(qB)$.

This allows precise measurement of isotopic masses and is a foundational tool in both chemistry and physics.