A disk of radius \( R \) with uniform positive charge density \( \sigma \) is placed on the \( x...
0A disk of radius \( R \) with uniform positive charge density \( \sigma \) is placed on the \( x y \) plane with its center at the origin. The Coulomb potential along the \( z \)-axis is
\[
V(z)=\frac{\sigma}{2 \epsilon_{0}}\left(\sqrt{R^{2}+z^{2}}-z\right)
\]
A particle of positive charge \( q \) is placed initially at rest at a point on the \( z \) axis with \( z=z_{0} \) and \( z_{0}0 \). In addition to the Coulomb force, the particle experiences a vertical force \( \vec{F}=-c \hat{k} \) with \( c0 \). Let \( \beta=\frac{2 c \epsilon_{0}}{q \sigma} \). Which of the following statements(s) is (are) correct?
(a) For \( \beta=\frac{1}{4} \) and \( z_{0}=\frac{25}{7} R \), the particle reaches the origin
(b) For \( \beta=\frac{1}{4} \) and \( z_{0}=\frac{3}{7} R \), the particle reaches the origin
(c) For \( \beta=\frac{1}{4} \) and \( z_{0}=\frac{R}{\sqrt{3}} \), the particle returns back to \( z=z_{0} \)
(d) For \( \beta1 \) and \( z_{0}0 \), the particle always reaches the origin
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