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Question

If a small sphere of mass m and charge q is hung 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 is

A
ϵ0(mgq)tan θ
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B
ϵ0(2mgq)tan θ
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C
ϵ0(mgq)tan θ
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D
ϵ0(mg3q)tan θ
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Solution

The correct option is B ϵ0(2mgq)tan θ
There will be two components of the tension.
Tcosθ=mg
Tsinθ=electric force
Tsinθ=qE
where E is the electric field due to the plate.
tan θ=qEmg
E=mg tan θqσϵ0=mg tan θqσ=mg ϵ0 tan θq
this is the surface charge density of the sphere

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