Direction of the force and trajectory

Problem

A positive charge enters with horizontal velocity $\vec v$ pointing rightwards into a region with magnetic field $\vec B$ pointing out of the page, as shown in the figure. In which direction is the magnetic force directed? After answering, draw the trajectory followed by the charge.

Solution

We apply the right-hand rule to the Lorentz force $\vec F = q\,\vec v \times \vec B$.

• Index finger in the direction of $\vec v$ (pointing right);
• middle finger in the direction of $\vec B$ (out of the page, towards us);
• the thumb then indicates the direction of $\vec v \times \vec B$, i.e. UPWARDS.

Since the charge is positive, the force has the same direction as the cross product: upwards.

The magnetic force is always perpendicular to the velocity, so it does not change the magnitude of $\vec v$ but continuously curves its direction: the charge traces an arc of a circle curving upwards (circular motion, clockwise as seen from the front).

$\ev{\vec F \text{ directed upwards}}$

Links

Topics: Magnetism Concepts: Lorentz force Skills: Right-hand rule