Question

A small sphere with mass m carries a positive charge q and is attached to one end of a silk fiber of length l.The other end of the fiber is attached to a large vertical insulating sheet that has a positive surface charge density row.When the sphere is in equilibrium at what angle the fiber makes with the vertical sheet

Answer 1

A small sphere of mass m and charge q is hung from silk thread at angle θ with the surface of a vertical charge conducting plate as shown in figure.

We know, Electric field intensity act due to plate = σ/ε₀, where σ is surface charge density of plate.
At equilibrium ,
Upward force = downward force
Tcosθ = mg ----(1)
Forward force = backward force
Tsinθ = σq/ε₀ [ ∵ F = qE , ∴ F = qσ/ε₀ ] -----(2)

Dividing equation (2) by (1)
Tsinθ/Tcosθ = qσ/ε₀mg
⇒Tanθ = qσ/ε₀mg
σ = ε₀mg tanθ/q

Hence, surface charge density of plate is σ = ε₀mg tanθ/q