JEE Advanced 2022 Paper 2 Q12 Physics Electrostatics & Circuits Electric Potential Medium

JEE Advanced 2022 Paper 2 · Q12 · Electric Potential

A disk of radius $R$ with uniform positive charge density $\sigma$ is placed on the $xy$ plane with its centre at the origin. The Coulomb potential along the $z$-axis is $V(z) = \dfrac{\sigma}{2\epsilon_0}\left(\sqrt{R^2 + z^2} - z\right).$ 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 $\beta = \dfrac{2 c \epsilon_0}{q\sigma}$. Which of the following statement(s) is(are) correct?

  1. A. For $\beta = 1/4$ and $z_0 = 25 R/7$, the particle reaches the origin.
  2. B. For $\beta = 1/4$ and $z_0 = 3 R/7$, the particle reaches the origin.
  3. C. For $\beta = 1/4$ and $z_0 = R/\sqrt{3}$, the particle returns back to $z = z_0$.
  4. D. For $\beta > 1$ and $z_0 > 0$, the particle always reaches the origin.
JEE Advanced multi-correct — pick every correct option, then check.
Reveal answer + step-by-step solution

Correct answer:A, C, D

Solution

The Coulomb force on the charge from the disk along $z$ is $F_C = -q\,dV/dz = (q\sigma/2\epsilon_0)(1 - z/\sqrt{R^2+z^2})$ (positive upward). The total force is $F_C - c$ (downward). Setting net force zero at $z = Z_0$: $1 - Z_0/\sqrt{R^2+Z_0^2} = 2c\epsilon_0/(q\sigma) = \beta$. For $\beta = 1/4$: $Z_0/\sqrt{R^2+Z_0^2} = 3/4$, $Z_0^2 = (9/16)(R^2+Z_0^2)$, $Z_0^2(7/16) = 9R^2/16$, $Z_0 = 3R/\sqrt{7} \approx 1.134\,R$. (A) $z_0 = 25R/7 \approx 3.57 R > Z_0$: above equilibrium, net force points down (toward origin); particle accelerates toward origin and reaches it — correct. (B) $z_0 = 3R/7 \approx 0.43 R < Z_0$: below equilibrium, Coulomb force exceeds gravity-like $c$; particle is pushed up, away from origin — does not reach origin — incorrect. (C) $z_0 = R/\sqrt{3} \approx 0.577 R < Z_0$: below equilibrium, particle is pushed up, oscillates and returns to $z_0$ (turning point) — correct. (D) $\beta > 1$: requires $1 - z/\sqrt{R^2+z^2} > 1$, impossible — so no equilibrium and Coulomb force is always less than $c$; net force always points to the origin, particle always reaches origin — correct. Official key: (A), (C), (D).

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