PX157 - D1e - a rotating conducting rod in a magnetic field

using cylindrical coordinates, $(R, ϕ, z)$
$\vec{B} = - B \hat{z}$
$\vec{v} = v_{ϕ} \hat{ϕ} = R ω \hat{ϕ}$
$ω$ is constant
choose path $C$ , radially outwards along the rod

\begin{aligned} \vec{v} \times \vec{B} \cdot d \vec{l} & = R ω \hat{ϕ} \times (- B) \hat{z} \cdot d R \hat{R} \\ = - R B ω d R \end{aligned}

\begin{aligned} ϵ & = \oint_{C} (\vec{v} \times \vec{B}) \cdot d \vec{l} \\ = - B ω \int_{0}^{L} R d R \\ = - \frac{1}{2} B ω L^{2} \end{aligned}

A (t) = \frac{1}{2} L^{2} ϕ (t)

\begin{aligned} ϵ & = - \frac{d}{d t} (B A) \\ = - B \frac{d A}{d t} \\ = - \frac{1}{2} B ω L^{2} \end{aligned}