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)