Question

A 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 $c>0$ Let $\beta=\frac{2 c \in_0}{q \sigma}$. Which of the following statement(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 $\beta>1$ and $z_0>0$, the particle always reaches the origin.

Answer 1

Answer: (A), (C), (D)

For $\beta=\frac{1}{4}$ and $z_0=\frac{25}{7} R$, the particle reaches the origin.

Answer 2

Given:
$F_1 = \frac{26\sqrt{R^2 + Z^2}}{2E_0}$
$F_2 = -ck$
$\frac{\alpha \sigma}{280}$​

For equilibrium at $z = \frac{Z}{o}$
$F_1 = F_2$

$\frac{\alpha \sigma}{280} \cdot \frac{Z}{\sqrt{R^2 + Z^2}} = c$

From equation (1):
$c = (1 - 3\frac{Z}{\sqrt{R^2 + Z^2}})$

$\frac{Z}{\sqrt{R^2 + Z^2}} = c(1 - 3\frac{Z}{\sqrt{R^2 + Z^2}})$

$\frac{1}{4} \cdot \frac{Z}{\sqrt{R^2 + Z^2}} = \frac{4c}{2c_{80}}$

$\frac{7}{\sqrt{R} - R} = 1.13R$

$𝑍>1.13𝑅⇒𝐹_2>𝐹_1$​ Particle reaches the origin.

$𝑍<1.13𝑅⇒𝐹_1>𝐹_2$ Particle reaches back to 𝑧=𝑍𝑜