An electron moves in a helical trajectory within a uniform magnetic field oriented along the y-axis. The position of the electron is described by the vector


$\begin{array}{l}\vec{r}=b\cos \left(\frac{2\pi y}{c}\right)\hat{\mathbf{i}}+y\ \hat{\mathbf{j}}+b\sin (\frac{2\pi y}{c})\hat{\mathbf{\ k}}\end{array}$

where $b$ represents the radius of the helical path, $c$ denotes the pitch of the helix, and $y$ varies as $y=v_{y}t$ , where $v_{y}$ is the constant component of velocity along the $y-axis$. Determine the time-dependent angular momentum $\vec{L}$ of the electron about the origin.

• A. $mbv_y\left[\frac{2\pi y}{c}\cos (\frac{2\pi y}{c})-\sin (\frac{2\pi y}{c})\hat{\mathbf{i}}-\frac{2\pi }{c}b\ \hat{\mathbf{j}}+\left(\cos (\frac{2\pi y}{c})+\frac{2\pi y}{c}\sin (\frac{2\pi y}{c})\right]\hat{\mathbf{k}}\right]$
• B. $\begin{array}{l}mbv_y\left[[\sin (\frac{2\pi y}{c})-\cos (\frac{2\pi y}{c})]\ \hat{\mathbf{i}}-\frac{2\pi b}{c}\ \hat{\mathbf{j}}+\left[\sin (\frac{2\pi y}{c})+\frac{2\pi y}{c}\cos (\frac{2\pi y}{c})\right]\ \hat{\mathbf{k}}\right]\end{array}$
• C. $mbv_y\left[\left[\cos \left(\frac{2\pi y}{c}\right)-\sin \left(\frac{2\pi y}{c}\right)\right]\hat{\mathbf{i}}+\left[\cos \left(\frac{2\pi y}{c}\right)+\frac{2\pi y}{c}\sin \left(\frac{2\pi y}{c}\right)\right]\hat{\mathbf{k}}\right]$
• D. $m v_{y} \frac{2 \pi b^{2}}{c} \hat{\mathbf{j}}$

Answer: (A)