Motion in a Magnetic Field

Figure 4.5 is a schematic diagram that illustrates the circular motion of a positively charged particle when it enters a uniform magnetic field perpendicular to its velocity.

The diagram shows a square region filled with cross symbols (×). Each cross represents the tail of an arrow, indicating that the magnetic field $\mathbf{B}$ is directed into the plane of the page (perpendicular to the page, away from the reader). A bold label "B" at the top reinforces this.

Inside this field region, a large circle is drawn. On this circle, several positive charges $+q$ are placed at different positions. At each charge, a tangent arrow shows the velocity $\mathbf{v}$ of the particle — the direction of motion at that instant. A radial arrow points from each charge directly inward toward the centre of the circle, representing the magnetic force $\mathbf{F}$ acting on the charge. This force is always perpendicular to both $\mathbf{v}$ and $\mathbf{B}$, and here it points toward the centre, acting as a centripetal force.

A radius $r$ is drawn from the centre to the circle, and a point P is marked at the lower-left part of the circle (likely to indicate a specific location for reference). The entire diagram is rendered in blue.

Physical idea

The figure teaches that when a charged particle moves perpendicular to a uniform magnetic field, the magnetic force $\mathbf{F} = q (\mathbf{v} \times \mathbf{B})$ is always perpendicular to the velocity. This force does no work (since it is perpendicular to displacement), so the speed of the particle remains constant, but the direction changes continuously. The force acts as a centripetal force, causing the particle to follow a circular path of constant radius.

Key formula derived from this figure

From the textbook, equating the magnetic force to the centripetal force gives:

Solving for the radius of the circular path:

where:

• $m$ = mass of the particle
• $v$ = speed (magnitude of velocity)
• $q$ = charge of the particle
• $B$ = magnitude of the uniform magnetic field …

What the figure shows

The diagram is a 3‑D sketch with the $y$-axis pointing upward, the $x$-axis to the right, and the $z$-axis slanting toward the lower left. A uniform magnetic field $\mathbf{B}$ points along the positive $x$-direction (a label “B” sits on the $x$-axis). A blue helix of several turns winds around the $x$-axis. At the left end of the helix a positive charge $+q$ is shown. Its velocity vector is split into two components: $\mathbf{v}_\perp$ (perpendicular to $\mathbf{B}$, drawn in the $y$–$z$ plane) and $\mathbf{v}_\parallel$ (parallel to $\mathbf{B}$, along the $x$-axis). Two labels appear below the helix: pitch (the axial distance the particle advances in one full turn) and radius (the radius of the circular part of the motion).

Physical idea

When a charged particle moves in a uniform magnetic field, the magnetic force $\mathbf{F} = q\,\mathbf{v} \times \mathbf{B}$ is always perpendicular to both $\mathbf{v}$ and $\mathbf{B}$. This force does no work, so the speed $|\mathbf{v}|$ stays constant. If the velocity has a component parallel to $\mathbf{B}$, that component ($v_\parallel$) is unaffected because the magnetic force has no component along $\mathbf{B}$. The perpendicular component ($v_\perp$) experiences a centripetal force that makes the particle go in a circle. The combination of uniform circular motion in the plane perpendicular to $\mathbf{B}$ and constant motion along $\mathbf{B}$ produces a helical path — the particle spirals forward along the field lines.

Key formulas developed from this figure

The radius of the circular part (the helix radius) comes from equating the magnetic force to the centripetal force:

where $m$ is the particle’s mass, $q$ its charge, $B$ the magnetic field strength, and $v_\perp$ the component of velocity perpendicular to $\mathbf{B}$.

The angular frequency (cyclotron frequency) is independent of speed: …