22.5 Force on a Moving Charge in a Magnetic Field: Examples and Applications

Magnetic Fields and Charged Particle Motion

When a charged particle moves through a magnetic field, the field pushes on it in a direction perpendicular to both its velocity and the field itself. This sideways push is what causes charged particles to curve into circular or spiral paths. The same principle is at work in particle accelerators, mass spectrometers, and even the auroras you see near Earth's poles.

Motion of Charged Particles in Magnetic Fields

The magnetic force on a moving charge is always perpendicular to the particle's velocity. Because the force never acts along the direction of motion, it can't speed the particle up or slow it down. Instead, it continuously deflects the particle sideways, bending its path into a curve.

Circular motion happens when the particle's velocity is exactly perpendicular to the magnetic field. The magnetic force acts as a centripetal force, pulling the particle into uniform circular motion at constant speed. Electrons curving through a cathode ray tube are a classic example.

Spiral (helical) motion happens when the velocity has some component parallel to the field. The parallel component is unaffected by the magnetic force, so the particle drifts steadily in that direction while simultaneously circling due to the perpendicular component. The result is a corkscrew-shaped path. The tighter or more stretched-out the spiral depends on the ratio of the parallel to perpendicular velocity components. Electrons trapped in Earth's magnetic field near the poles follow helical paths like this, which is what produces auroras.

To find the direction of the magnetic force, use the right-hand rule:

1. Point your fingers in the direction of the particle's velocity ($v$).
2. Curl them toward the direction of the magnetic field ($B$).
3. Your thumb points in the direction of the force on a positive charge.
4. For a negative charge, the force points opposite your thumb.

Radius of Curvature Calculations

The radius of the circular path is given by:

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

where $m$ is the particle's mass, $v$ is the component of velocity perpendicular to the field, $q$ is the particle's charge, and $B$ is the magnetic field strength.

This formula connects directly to circular motion. The magnetic force provides the centripetal force: $qvB = \frac{mv^2}{r}$. Solving for $r$ gives the equation above.

A few proportionalities worth remembering:

• Larger mass or higher speed → larger radius. The particle is harder to deflect, so it curves more gently. A proton curves much more widely than an electron at the same speed because the proton is about 1,836 times more massive.
• Larger charge or stronger field → smaller radius. The force is stronger, so the particle curves more tightly. An alpha particle (charge $2e$) curves more tightly than a proton (charge $e$) at the same speed and mass-per-charge ratio.

Example: Suppose a proton ($m = 1.67 \times 10^{-27}$ kg, $q = 1.6 \times 10^{-19}$ C) moves at $3.0 \times 10^{6}$ m/s perpendicular to a 0.50 T magnetic field.

1. Plug into the formula: $r = \frac{(1.67 \times 10^{-27})(3.0 \times 10^{6})}{(1.6 \times 10^{-19})(0.50)}$
2. Numerator: $5.01 \times 10^{-21}$
3. Denominator: $8.0 \times 10^{-20}$
4. $r = 0.063$ m, or about 6.3 cm

Applications of Magnetic Forces

Particle accelerators use magnetic fields to steer charged particles along controlled paths while electric fields boost their energy.

• Linear accelerators (like SLAC at Stanford) accelerate particles in a straight line using alternating electric fields. Magnets are used for focusing, not for bending into a circular path.
• Cyclotrons use a constant magnetic field to bend particles into a spiral while an alternating electric field accelerates them each time they cross a gap between two D-shaped electrodes. These are widely used for producing medical isotopes.
• Synchrotrons (like the Large Hadron Collider) keep particles on a fixed circular path by increasing the magnetic field strength in sync with the particles' rising energy. This avoids the need for an enormous single magnet.

Mass spectrometers separate ions by their mass-to-charge ratio using the radius of curvature relationship. Here's the basic process:

1. Atoms or molecules are ionized (given a charge).
2. The ions are accelerated through a known electric potential to give them a known kinetic energy.
3. The ions enter a uniform magnetic field, where they curve into circular arcs.
4. Because $r = \frac{mv}{qB}$, ions with different mass-to-charge ratios follow arcs of different radii and land at different positions on a detector.

This technique is used in drug testing to identify chemical compounds, in carbon dating to measure isotope ratios, and in biochemistry to analyze protein structures.

Additional Concepts in Electromagnetic Interactions

• The magnetic force on a moving charge is mathematically described by the cross product: $\vec{F} = q\vec{v} \times \vec{B}$. The magnitude is $F = qvB\sin\theta$, where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$. When $\theta = 0°$ (particle moves parallel to the field), the force is zero.
• Magnetic flux density ($B$) describes the strength and direction of a magnetic field at a point in space. Its SI unit is the tesla (T).
• The Hall effect occurs when a magnetic field is applied perpendicular to a current-carrying conductor. The field deflects the moving charges to one side, creating a measurable voltage across the conductor. This voltage can be used to determine the sign of the charge carriers and to measure magnetic field strength.