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

θ = arctan(σq/2εmg)

Explanation:

(See attachment below)

The electric field of the infinite sheet is given by

E = σ/2ε -------- Equation 1

The force acting on the ball is then given by:

F = Eq = σq/2ε ------ Equation 2

The electric force is equal to the tension component

The tension component is given by Tsin(θ)

Tsin(θ) = σq/2ε ---------- Equation 3

The vertical component is given by

Tcos(θ)

And the vertical component is equal to the weight

So, Tcos(θ) = mg -----_ Equation 4

Divide Equation 3 by 4, to give

Tsin(θ) / Tcos(θ) = σq/2ε ÷ mg

sin(θ) / cos(θ) = σq/2ε * 1/mg

tan(θ) = σq/2εmg

θ = arctan(σq/2εmg)