PX157 - C1c - a charge in a magnetic field

the motion of a charge in a magnetic field

a charge, $q$ , in a uniform magnetic field, $\vec{B}$
splitting the velocity vector of the charge, $\vec{v}$ , into two components:

\vec{v} (t) = {\vec{v}}_{∥} + {\vec{v}}_{⊥}

${\vec{v}}_{∥} \times \vec{B} = 0$ and ${\vec{v}}_{⊥} \cdot \vec{B} = 0$

\begin{aligned} \vec{F} & = q \vec{v} \times \vec{B} \\ = q {\vec{v}}_{⊥} \times \vec{B} \\ ∴ | \vec{F} | & = | q | v_{⊥} B \end{aligned}

balancing this force against the centripetal acceleration:

\begin{aligned} | q | v_{⊥} B & = \frac{m v_{⊥}^{2}}{R_{L}} \\ ∴ R_{L} & = \frac{m v_{⊥}}{| q | B} \end{aligned}

where, $R_{L} =$ larmer radius

parallel motion has no force as ${\vec{v}}_{∥} \times \vec{B} = \vec{0}$ , $∴ {\vec{v}}_{∥}$ is constant

the motion is a helix
note: to figure the direction of force, use left hand thumb rule for positive charge and right hand thumb rule for negative charge

work done by moving charge

equation of motion:

\begin{aligned} m \frac{d \vec{v}}{d t} & = q \vec{v} \times \vec{B} \\ m \frac{d \vec{v}}{d t} \cdot \vec{v} & = q (\vec{v} \times \vec{B}) \cdot \vec{v} \\ \frac{1}{2} m \frac{d}{d t} (\vec{v} \cdot \vec{v}) & = 0 \\ \frac{1}{2} m \frac{d}{d t} v^{2} & = 0 \\ \frac{d}{d t} (\frac{1}{2} m v^{2}) & = 0 \end{aligned}

$K E$ is constant $⟹$ no work done
therefore, a magnetic field (constant in time and space) does no work
alternatively,

W = \int q (\vec{v} \times \vec{B}) \cdot d \vec{l}

$d \vec{l}$ is along the path, but $\vec{v}$ is also along the path:

\vec{v} = \frac{d \vec{l}}{d t} ⟹ \vec{v} ∥ d \vec{l} ⟹ W = 0