Question 705059

Force $\vec{F} = q[\vec{v} \times \vec{B}]$. Torque $\vec{\tau} = \vec{r} \times \vec{F} = \vec{r} \times (\vec{v} \times \vec{B}) = \vec{v}(\vec{r} \cdot \vec{B}) - \vec{B}(\vec{r} \cdot \vec{v})$. Given $\vec{B} = b_0 \frac{\vec{r}}{r^3}$, $\vec{r} \cdot \vec{B} = 0$, so $\vec{\tau} = -\vec{B}(\vec{r} \cdot \vec{v}) = -b_0 \frac{\vec{r}}{r^3} \frac{dr}{dt} = b_0 \frac{d}{dt} (\frac{\vec{r}}{r})$. Since $\vec{\tau} = d\vec{L}/dt$, the result follows.