\(F = qE = \frac {qp}{4\pi \varepsilon_0 r^3} (2 \cos \theta \hat r + \sin \theta \hat \theta)\)
Force is radially inward
\(\hat r = \cos \theta \hat z + \sin \theta \hat S\)
\(\hat \theta =- \sin \theta \hat z +\cos \theta \hat S\)
\(F =\frac{qp}{4\pi \varepsilon _0r^3} ((3\cos ^2\theta - 1)\hat z) + 3\sin \theta\cos \theta \hat s\)
The force needs to be purely radial and inward and negative
\(\cos \theta = \frac 1{\sqrt 3}\)
Using \(S = r \sin \theta\). centripetal force
\(F_{cp} = qE(S_q\cos \theta = \frac 1{\sqrt 3}) \)
\(= \frac{-qP}{\pi \varepsilon_0 S^2 3^{3/2}}\)
\(F_{cp} = \frac{-mv^2}S\)
⇒ \(v = \sqrt{\frac{-F_{cp}S}{m}}\)
\(= \sqrt{\frac{qp}{\pi \varepsilon_0S^2 3^{3/2}m}}\)
Angular momentum
\(L = mS^2\omega\)
\(= mvS\)
\(= \sqrt{\frac{qpm}{\pi \varepsilon_0 3^{3/2}}}\)
\(E = T + v\)
\(= qv + \frac 12mv^2\)
\(= \frac{-qp}{4\pi \varepsilon_0 S^2 \sqrt 3} (1 - \frac 13) + \frac 12 \frac {qp}{\pi \varepsilon_0S^2 3^ {1/2}}\)
