Physics /Electromagnetism /Module 1.1

Magnetic Force on Moving Charges

Master the 3D mechanics of the Lorentz Force and the Right-Hand Rule through high-fidelity interactive spatial simulations.

When a charged particle enters a magnetic field, it experiences a force. But unlike gravity or electric forces, which pull straight toward or push straight away, the magnetic force operates exclusively in the third dimension.

The Cross Product in Reality

The interaction between a moving charge and a magnetic field is defined by the Lorentz Force equation. Because this requires a mathematical cross product , the resulting force is always perfectly perpendicular to both the particle's velocity vector $\vec{v}$ and the magnetic field vector $\vec{B}$ .

Visualizing Vector Directions

Because visualizing 3D vectors on a 2D piece of paper is incredibly difficult, physicists use the Right-Hand Rule. Point your fingers in the direction of velocity, curl them toward the magnetic field, and your thumb points toward the resulting force.

Uniform Circular Motion

Because the magnetic force is always perpendicular to velocity, it cannot do work on the particle. It changes the particle's direction, but never its speed.

When a negative charge (such as an electron) moves perpendicular to a uniform magnetic field, this constant perpendicular force acts as a centripetal force , locking the particle into perfect circular motion.

If the velocity enters at an angle, the parallel component is unaffected, stretching the orbit into a helical path.

The Mathematics of the Orbit

Because the magnetic force acts as a centripetal force ( $F_B = F_c$ ), we can set the Lorentz force equal to the centripetal force equation to find the exact radius ( $r$ ) of the particle's orbit:

qvB = \frac{mv^2}{r} \implies r = \frac{mv}{qB}

Notice that the radius depends on the particle's momentum ( $mv$ ). Faster or heavier particles make wider circles. Stronger magnetic fields make tighter circles.

We can also calculate the Period ( $T$ )—the time it takes to complete one full orbit. Since Time = Distance / Speed ( $T = \frac{2\pi r}{v}$ ), we can substitute our radius formula to get:

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

Notice something fascinating: The period is completely independent of velocity. A faster particle travels a wider circle, but it completes the orbit in the exact same amount of time as a slower particle!

Concept Checks

The “Parallel” Trap

A proton is fired at a high speed directly East. Sensors detect absolutely zero magnetic force acting on the particle. Can you definitively conclude that there is no magnetic field in this region?

The Electron Reversal

An electron is fired directly into your screen (into the page). The background magnetic field points straight down. In which direction will the electron be deflected?

The Circular Orbit

A proton enters a

1.5 \text{ T}

uniform magnetic field at a perfect

90^\circ

angle, traveling at

3.0 \times 10^6 \text{ m/s}

. What is the radius of the circular path it gets locked into? (Mass of proton = $1.67 \times 10^{-27} \text{ kg}$ , Charge = $1.6 \times 10^{-19} \text{ C}$ )