An infinite, vertical, nonconducting plane sheet is uniformly charged with electricity. Next to the sheet is a dipole that can freely oscillate about its midpoint O, which is at a distance Ж from the sheet. Each end of the dipole bears a charge q and a mass m. The length of the dipole is 2L. When set into small oscillations, it oscillates with a frequency of ν Hertz. What is the surface charge density σ on the plane? Express your answer in terms of some or all of the variables q, m, L, Ж, v, ε.
τ = p x E
Iα = -pEsinθ = -pEθ (for small oscillations)
T = 2π √(I/pE)
E = σ/2ε (for infinite sheet/plane)
- τ = torque
- I = rotational inertia
- p = dipole moment
- E = electric field
- σ = surface charge density
The Attempt at a Solution
The surface charge density σ of the plane is σ = 2εE, so the electric field of the sheet must be found. I think at some point the moment of inertia will be needed; do you treat the dipole as two spheres on a massless rod? How would one go about calculating this?
If v is the frequency, then
1/v = 2π √(I/pE)
The dipole moment is 2qL. Solving for E..
E = 2qL/I(2πv)²
Will this E be the magnitude of the field acting on the dipole? Will this be the field of the sheet? And how does the distance of the dipole from the sheet (Ж) come into play? I recall my prof telling us that the field of an infinite sheet on a point charge was independent of the distance away from it (hence, why E = σ/2ε). Does it come into play in this situation?