JEE Advanced 2022 · Paper 2 · Physics Q12

← JEE Advanced 2022 · Paper 2

PhysicsMultiple correct+4 / -2Question 30 of 54

A disk of radius $R$ with uniform positive charge density $\sigma$ is placed on the $xy$ 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 $\beta = \dfrac{2c\epsilon_0}{q\sigma}$ . Which of the following statement(s) is(are) correct?

For

\beta = \frac14

and

z_0 = \frac{25}{7}R

, the particle reaches the origin.

For

\beta = \frac14

and

z_0 = \frac37R

, the particle reaches the origin.

For

\beta = \frac14

and

z_0 = \frac{R}{\sqrt3}

, the particle returns back to

z = z_0

For

\beta > 1

and

z_0 > 0

, the particle always reaches the origin.

Answer and worked solution

Official answer: A, C, D

Potential energy. In units of $\frac{q\sigma}{2\epsilon_0}$ :
$u(z)=\sqrt{R^2+z^2}-z+\beta z,\qquad u'(z)=\frac z{\sqrt{R^2+z^2}}-1+\beta.$
Starting from rest at $z_0$ , the particle moves toward lower $u$ .

Case $\beta=\frac14$ . $u'=0$ at $\frac z{\sqrt{R^2+z^2}}=\frac34$ , i.e. $z^*=\frac{3R}{\sqrt7}\approx1.13R$ . This is a minimum of $u$ .

Starting above $z^*$ , the particle moves down. It reaches the origin if $u(z_0)\ge u(0)=R$ .
Starting below $z^*$ , it is pushed up and oscillates about $z^*$ , returning to $z_0$ .

Check the options.

(A) $z_0=\frac{25}7R$ : $u=R\left(\frac{\sqrt{674}}7-\frac{25}{28}\right)\approx1.03R\ge R$ . It reaches the origin. True.
(B) $z_0=\frac37R<z^*$ : it moves away from the disk. False.
(C) $z_0=\frac R{\sqrt3}<z^*$ : it oscillates and returns to $z_0$ . True.
(D) $\beta>1$ : $u'>0$ for every $z$ , so the net force always points toward the disk, and the particle reaches the origin. True.

(A), (C), (D).

Topics:Electric potential Conservation of mechanical energy

← Previous question Next question →

JEE Advanced 2022 · Paper 2, Physics question 12