Finding the surface charge density of an infinite sheet next to a dipole

[mclame22]

Homework Statement

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.

Homework Equations

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?

 

[mark726]

To solve for the surface charge density σ, we first need to find the electric field E at the location of the dipole. We can use the equation E = σ/2ε for an infinite sheet, where σ is the surface charge density and ε is the permittivity of the medium.

Next, we need to find the torque (τ) acting on the dipole due to the electric field. We can use the equation τ = p x E, where p is the dipole moment and E is the electric field.

We also know that for small oscillations, the torque is equal to the product of the rotational inertia (I) and the angular acceleration (α). So we can write the equation τ = Iα.

Combining these equations, we get:

Iα = p x E = pEsinθ

Since the oscillations are small, we can approximate sinθ as θ. So we have:

Iα = pEθ

Next, we can use the equation T = 2π√(I/pE) to find the period of oscillation (T) of the dipole.

Finally, we can use the given frequency (ν) to find the angular velocity (ω) of the dipole, using the equation ν = ω/2π.

Now, we can substitute the values of I, α, p, E, θ, T and ω into the equation Iα = pEθ to solve for the surface charge density σ.

σ = 2εE = 2ε(p/I)α = 2ε(p/I)ω²θ = 2ε(p/I)(ν/2π)²θ

Substituting the values of p, I, ν, and θ, we get:

σ = 2ε(2qL/m)(ν/2π)²(2Lcosθ) = 4εqL²ν²cosθ/mπ²

Therefore, the surface charge density σ is given by:

σ = 4εqL²ν²cosθ/mπ²

where q is the charge on each end of the dipole, L is the length of the dipole, ν is the frequency of oscillation, θ is the angle between the dipole moment and the electric field, m is the mass of each end of the dipole, and ε is the permittivity of the medium.