A disk of radius with uniform positive charge density is placed on the -plane with its center at the origin. The Coulomb potential along the -axis is A particle of positive charge is placed initially at rest at a point on the -axis with and . In add

Question

A disk of radius $R$ with uniform positive charge density $σ$ is placed on the $x y$ -plane with its center at the origin.

The Coulomb potential along the $z$ -axis is $V (z) = \frac{σ}{2 ε_{0}} (\sqrt{R^{2} + z^{2} - z}) .$ 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 $β = \frac{2 c ε_{0}}{q σ} .$ Which of the following statement(s) is(are) correct? [2022]

Smart Achievers · Accepted Answer

(1, 3, 4)

Particle will reach the origin only if

$Δ K \geq 0$

$\Rightarrow W_{el} + W_{ext} > 0$

$\Rightarrow \frac{σ q}{2 ε_{0}} [(\sqrt{R^{2} + 0^{2}} - 0) + (\sqrt{R^{2} + Z_{0}^{2}} - Z_{0})] + C Z_{0} \geq 0$

$\Rightarrow \frac{σ q}{2 ε_{0}} (R - \sqrt{R^{2} + Z_{0}^{2}} + Z_{0}) + C Z_{0} \geq 0$

$\Rightarrow \frac{σ q}{2 ε_{0}} C (R - \sqrt{R^{2} + Z_{0}^{2}} + Z_{0}) + \frac{C Z_{0}}{C} \geq 0$

$\Rightarrow \frac{1}{β} (R + Z_{0} - \sqrt{R^{2} + Z_{0}^{2}}) + Z_{0} \geq 0$

Substitute $Z_{0}$ and $β$ , and check the condition.

If $Δ K > 0$ , the particle will reach the origin; otherwise, it will not reach the origin.

Solution

1 For $β = \frac{1}{4}$ and $z_{0} = \frac{25}{7} R,$ the particle reaches the origin.

2 For $β = \frac{1}{4}$ and $z_{0} = \frac{3}{7} R,$ the particle reaches the origin.

3 For $β = \frac{1}{4}$ and $z_{0} = \frac{R}{\sqrt{3}},$ the particle returns back to $z = z_{0} .$

4 For $β > 1$ and $z_{0} > 0$ , the particle always reaches the origin.

Want Us to Email you About Special Offers & Updates?

Solution

1 For β=14 and z0=257R, the particle reaches the origin.

2 For β=14 and z0=37R, the particle reaches the origin.

3 For β=14 and z0=R3, the particle returns back to z=z0.

4 For β>1 and z0>0, the particle always reaches the origin.

Want Us to Email you About Special Offers & Updates?

1 For $β = \frac{1}{4}$ and $z_{0} = \frac{25}{7} R,$ the particle reaches the origin.

2 For $β = \frac{1}{4}$ and $z_{0} = \frac{3}{7} R,$ the particle reaches the origin.

3 For $β = \frac{1}{4}$ and $z_{0} = \frac{R}{\sqrt{3}},$ the particle returns back to $z = z_{0} .$

4 For $β > 1$ and $z_{0} > 0$ , the particle always reaches the origin.