Question

A particle having mass $m$ and charge $(-q)$ moves along an ellipse around a fixed charge $Q$ such that its maximum and minimum distances from the fixed charge are $r_{1}$ and $r_{2}$ respectively. Show that the angular momentum $L$ of this particle about $Q$ is a $\times \cdot 10^{4} Ns$, then find $a$. $\left(Q=2 \,C, q=1 \,C, r_{1}=3 \,m\right.$, $r _{2}=1 \,m , m =0.3 \,kg$ )

Answer 1

(1) Angular momentum conservation about charge $+Q$
$L = mV _{1} r _{1}= mV _{2} r _{2}\,\,\,...(1)$
(2) Energy conservation
$\frac{1}{2} mV _{1}{ }^{2}-\frac{ qQ }{4 \pi \varepsilon_{0} r _{1}}=\frac{1}{2} mV _{2}{ }^{2}-\frac{ qQ }{4 \pi \varepsilon_{0} r _{2}}\,\,\,\,...(2)$
from (1) and (2)
$V _{1}=\sqrt{\frac{ r _{2} Qq }{2 \pi \varepsilon_{0} m \left( r _{1}+ r _{2}\right) r _{1}}}$
(3) $L = mV _{1} r _{1}$
$L =\sqrt{\frac{ mr _{1} r _{2} Qq }{2 \pi \varepsilon_{0}\left( r _{1}+ r _{2}\right)}}$