If a small sphere of mass m and charge q is hanged from a silk thread at an angle θ with the surface of a vertical charged conducting plate, then for equilibrium of sphere the surface charge density of the plate -

Question

If a small sphere of mass m and charge q is hanged from a silk thread at an angle θ with the surface of a vertical charged conducting plate, then for equilibrium of sphere the surface charge density of the plate - (A) (ϵ 0 m g t a n θ /q) (B) (2 ϵ 0 m g t a n θ /q) (C) ϵ 0mgqtanθ  (D) (ϵ 0 m g t a n θ /3 q)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="Solution" src="https://cdn.tardigrade.in/q/nta/p-hwoezkuodcxqaj6j.jpg" /> T sin θ = qE ldots (.1.) T cos θ = mg ldots (.2.) tan θ = (q E/m g) = (q σ /ϵ 0 m g) σ = (ϵ 0 m g t a n θ /q)

Q. If a small sphere of mass $m$ and charge $q$ is hanged from a silk thread at an angle $θ$ with the surface of a vertical charged conducting plate, then for equilibrium of sphere the surface charge density of the plate -

$\frac{ϵ _{0} m g t an θ}{q}$

$\frac{2 ϵ _{0} m g t an θ}{q}$

$ϵ_{0} m g qt an θ$

$\frac{ϵ _{0} m g t an θ}{3 q}$

$T s in θ = qE \dots (1)$
$T cos θ = m g \dots (2)$
$t an θ = \frac{q E}{m g} = \frac{q σ}{ϵ _{0} m g}$
$σ = \frac{ϵ _{0} m g t an θ}{q}$

Q. If a small sphere of mass m and charge q is hanged from a silk thread at an angle θ with the surface of a vertical charged conducting plate, then for equilibrium of sphere the surface charge density of the plate -

qϵ0​mgtanθ​

q2ϵ0​mgtanθ​

ϵ0​mgqtanθ

3qϵ0​mgtanθ​