Ranking the Lorentz force (four directions)

Problem

Four equal charges $q$ move with the same $v_0$ in the east direction: (A) $\vec B$ to the north, magnitude $B$; (B) $\vec B$ to the east, magnitude $2B$; (C) $\vec B$ vertical, magnitude $B$; (D) $\vec B$ at $45^\circ$ east-north, magnitude $B$. Rank from the largest to the smallest force magnitude.

Solution

The magnitude of the Lorentz force depends on the angle $\theta$ between $\vec v$ (east) and $\vec B$: $F = q v_0 B \sin\theta$

• (A) $\vec B$ to the north is perpendicular to east: $\sin 90° = 1$, so $F \propto B$.
• (B) $\vec B$ to the east is parallel to $\vec v$: $\sin 0° = 0$, so $F = 0$ (irrelevant that the magnitude is $2B$).
• (C) $\vec B$ vertical is perpendicular to east: $\sin 90° = 1$, so $F \propto B$.
• (D) $\vec B$ at $45°$: $\sin 45° \approx 0{,}71$, so $F \propto 0{,}71\,B$.

Comparing: A and C are both worth $1{,}0\,B$, D is worth $0{,}71\,B$, B is worth $0$. $\ev{A = C > D > B}$

Links

Topics: Magnetismo Concepts: Forza di Lorentz Skills: Ragionamento di ranking