Physics Problem: Electric Field and Tension Angle
Question
A small sphere of mass 0.25 g (i.e. 0.25 × 10-3 kg) carries a charge of 9.0 × 10−10 C. The sphere is attached to one end of a very thin silk string of length 5.0 cm. The other end of the string is connected to a large vertical conducting plate with a surface charge density of σ = 30 × 10−6 C/m².
What is the angle that the string makes with the vertical?
Solution
The sphere is in equilibrium under the action of three forces:
- Gravitational force downward: F_g = mg
- Electric force horizontally from the charged plate: F_e = qE
- Tension in the string, acting along the string
Step 1: Calculate the Electric Field
The electric field near a large conducting plate is:
E = σ / (2ε₀)
where ε₀ = 8.854 × 10−12 C²/N·m²
Step 2: Write Force Balance Equations
From equilibrium conditions:
Horizontal: T·sinθ = F_e
Vertical: T·cosθ = mg
Dividing these equations gives:
tanθ = F_e / mg
Step 3: Substitute and Solve
Using:
F_e = q·E = q·(σ / 2ε₀)
Substitute values:
tanθ = (q·σ) / (2·ε₀·m·g)
= (9 × 10−10 × 30 × 10−6) / (2 × 8.854 × 10−12 × 0.25 × 10−3 × 9.8)
Calculating this:
tanθ = 0.622
Step 4: Find the Angle
Taking inverse tangent:
θ = tan−1(0.622)
θ = 31.89°