Source: High school physics (Chinese)
Problem
The figure shows four configurations of a charged particle moving with velocity $\vec{v}$ in a uniform magnetic field $\vec{B}$. A dot $(\cdot)$ in the figure represents a vector directed out of the page. Determine the direction of the magnetic force on the particle in each case.
- Charge $+q$ moves upward in the plane of the page; $\vec{B}$ is directed out of the page.
- Charge $+q$ moves to the right in the plane of the page; $\vec{B}$ is directed downward in the plane of the page.
- Charge $+q$ moves in the plane of the page with both rightward and downward components, making an angle with $\vec{B}$; $\vec{B}$ is directed to the right in the plane of the page.
- Charge $-q$ moves out of the page; $\vec{B}$ is directed upward in the plane of the page.
Q1: rightward (in the plane of the page). Q2: into the page. Q3: out of the page. Q4: rightward (in the plane of the page).
Use $\vec{F} = q\,\vec{v}\times\vec{B}$ and the right-hand rule (reverse for a negative charge).
Q1: $\vec{v}$ up, $\vec{B}$ out of page. Right-hand rule: $\vec{v}\times\vec{B}$ points to the right; with $+q$, $\vec{F}$ is to the right (in the plane).
Q2: $\vec{v}$ right, $\vec{B}$ down. $\vec{v}\times\vec{B}$ points into the page; with $+q$, $\vec{F}$ is into the page.
Q3: Both $\vec{v}$ and $\vec{B}$ lie in the plane, so $\vec{F}$ is perpendicular to the plane. With $\vec{v}$ having rightward and downward components and $\vec{B}$ rightward, $\vec{v}\times\vec{B}$ points out of the page; with $+q$, $\vec{F}$ is out of the page.
Q4: $\vec{v}$ out of page, $\vec{B}$ up. $\vec{v}\times\vec{B}$ points to the left; with $-q$ the force is reversed, so $\vec{F}$ is to the right (in the plane).