Source: High school physics (Chinese)
Problem
At a location in the Northern Hemisphere, the Earth's magnetic field has magnitude $B = 0.6\times 10^{-4}$ T, magnetic declination of $0$, and magnetic inclination (dip angle) of $70^\circ$. Find the magnitude and direction of the magnetic force on an electron moving at $v = 2.0\times 10^{6}$ m/s in each of the following directions:
- Velocity directed vertically downward.
- Velocity directed from south to north.
- Velocity directed from west to east.
Q1: $F_1 \approx 6.6\times 10^{-18}$ N, directed west. Q2: $F_2 \approx 1.80\times 10^{-17}$ N, directed east. Q3: $F_3 \approx 1.92\times 10^{-17}$ N, in the magnetic-meridian plane, tilted $20^\circ$ below horizontal pointing south (perpendicular to $\vec{B}$).
With zero declination, $\vec{B}$ lies in the magnetic-meridian (north-vertical) plane, tilted $70^\circ$ below horizontal toward the north. Decompose with $\hat{x}$ east, $\hat{y}$ north, $\hat{z}$ up:
$$\vec{B} = B\cos 70^\circ\,\hat{y} - B\sin 70^\circ\,\hat{z}.$$For an electron, $q = -e$ and $\vec{F} = -e\,\vec{v}\times\vec{B}$. The reference product is $evB = (1.6\times 10^{-19})(2.0\times 10^{6})(0.6\times 10^{-4}) = 1.92\times 10^{-17}$ N.
Q1: $\vec{v} = -v\,\hat{z}$ (down). $\vec{v}\times\vec{B} = vB\cos 70^\circ\,\hat{x}$ (east). $\vec{F} = -evB\cos 70^\circ\,\hat{x}$ — west. Magnitude $F_1 = evB\cos 70^\circ \approx 1.92\times 10^{-17}\times 0.342 \approx 6.6\times 10^{-18}$ N.
Q2: $\vec{v} = v\,\hat{y}$ (north). $\vec{v}\times\vec{B} = -vB\sin 70^\circ\,\hat{x}$ (west). $\vec{F} = evB\sin 70^\circ\,\hat{x}$ — east. Magnitude $F_2 = evB\sin 70^\circ \approx 1.92\times 10^{-17}\times 0.940 \approx 1.80\times 10^{-17}$ N.
Q3: $\vec{v} = v\,\hat{x}$ (east), perpendicular to $\vec{B}$. $\vec{v}\times\vec{B} = vB\sin 70^\circ\,\hat{y} + vB\cos 70^\circ\,\hat{z}$ (north and up). $\vec{F} = -evB\sin 70^\circ\,\hat{y} - evB\cos 70^\circ\,\hat{z}$ — south and down. Magnitude $F_3 = evB \approx 1.92\times 10^{-17}$ N. Direction: perpendicular to $\vec{B}$ in the meridian plane, tilted $20^\circ$ below horizontal pointing south (equivalently, $70^\circ$ from the downward vertical, on the south side).