A small bob of mass 100 mg and charge +10 μC is connected to an insulating string of length 1 m and is brought near an infinitely large non-conducting sheet of charge density σ as shown in the original figure. If the string makes an angle 45° with the sheet at equilibrium, find σ. Given ε₀ = 8.85×10⁻¹² F m⁻¹ and g = 10 m s⁻².

Question

G1 Physics · Accepted Answer

For an infinite non-conducting sheet, E = σ/(2ε₀). The electric force on the bob is horizontal: Fₑ = qE = qσ/(2ε₀). At equilibrium, since the string makes 45° with the vertical sheet, tan45° = Fₑ/(mg) = 1. Therefore qσ/(2ε₀) = mg. So σ = 2ε₀mg/q. Convert values: m = 100 mg = 1.0×10⁻⁴ kg, q = 10 μC = 1.0×10⁻⁵ C. σ = [2(8.85×10⁻¹²)(1.0×10⁻⁴)(10)]/(1.0×10⁻⁵) = 1.77×10⁻⁹ C/m² = 1.77 nC/m². Correct option: B.