Ranking the Lorentz force (four angles)

Problem

Four charges $q = +1\;\mu\text{C}$ move in a uniform magnetic field $B$ pointing out of the page, with velocities forming different angles with $\vec B$: (A) $v = 100$ m/s, angle $30^\circ$, $B = 0{,}5$ T; (B) $v = 200$ m/s, angle $90^\circ$, $B = 0{,}5$ T; (C) $v = 100$ m/s, angle $0^\circ$, $B = 1$ T; (D) $v = 100$ m/s, angle $90^\circ$, $B = 0{,}5$ T. Rank the magnitudes of the forces $F = |q|vB\sin\theta$ from smallest to largest.

Solution

Since $|q|$ is the same for all of them, it is enough to compare the quantity $vB\sin\theta$: $\begin{aligned} A &= 100\cdot 0{,}5\cdot\sin 30^\circ = 100\cdot 0{,}5\cdot 0{,}5 = 25 \\ B &= 200\cdot 0{,}5\cdot\sin 90^\circ = 200\cdot 0{,}5\cdot 1 = 100 \\ C &= 100\cdot 1\cdot\sin 0^\circ = 0 \\ D &= 100\cdot 0{,}5\cdot\sin 90^\circ = 100\cdot 0{,}5\cdot 1 = 50 \end{aligned}$ Ranking in increasing order: $C(0) < A(25) < D(50) < B(100)$. $\ev{C < A < D < B}$

Links

Topics: Magnetism Concepts: Lorentz force Skills: Ranking reasoning