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Electrostatics question

2022 · Shift 2 · Q48
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Electrostatics question

2022 · Shift 2 · Q48

JEE AdvancedPhysicsElectrostaticsMultiple correct+4 / −2
A disk of radius R\mathrm{R}R with uniform positive charge density σ\sigmaσ is placed on the xyx yxy plane with its center at the origin. The Coulomb potential along the zzz-axis is V(z)=σ2ϵ0(R2+z2−z).V(z)=\frac{\sigma}{2 \epsilon_{0}}\left(\sqrt{R^{2}+z^{2}}-z\right) .V(z)=2ϵ0​σ​(R2+z2​−z). A particle of positive charge qqq is placed initially at rest at a point on the zzz axis with z=z0z=z_{0}z=z0​ and z0>0z_{0}\gt 0z0​>0. In addition to the Coulomb force, the particle experiences a vertical force F⃗=−ck^\vec{F}=-c \hat{k}F=−ck^ with c>0c\gt 0c>0. Let β=2cϵ0qσ\beta=\frac{2 c \epsilon_{0}}{q \sigma}β=qσ2cϵ0​​. Which of the following statement(s) is(are) correct?
  1. A
    For β=14\beta=\frac{1}{4}β=41​ and z0=257Rz_{0}=\frac{25}{7} Rz0​=725​R, the particle reaches the origin.
  2. B
    For β=14\beta=\frac{1}{4}β=41​ and z0=37Rz_{0}=\frac{3}{7} Rz0​=73​R, the particle reaches the origin.
  3. C
    For β=14\beta=\frac{1}{4}β=41​ and z0=R3z_{0}=\frac{R}{\sqrt{3}}z0​=3​R​, the particle returns back to z=z0z=z_{0}z=z0​.
  4. D
    For β>1\beta\gt 1β>1 and z0>0z_{0}\gt 0z0​>0, the particle always reaches the origin.
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