Question

Time period of a oscillatory dipole

Answer 1

There is a specific formula for the dipole moment and torque but we can sorta derive that as we go along:



We start with Newton's 2nd Law for circular motion: τ=Iα    ⋆

We place the origin at the mid point and exploit the symmetry and for minimum time we arrange the positive and negative charges as shown. The torque is:

τ=2×(r×F)

r=(L2cosθL2sinθ) and F=(Eq0)

⇒τ=−LEqsinθ ez

The inertia for 2 point masses is 2×M(L2)2=ML22

And so from ⋆:

−LEqsinθ ez=ML22..θ ez

We make the usual linearisation assumption, θ≈sinθ for small θ and we get this DE:

..θ+2EqMLθ=0 which is in usual form: ..θ+ω2θ=0

So period T=2πω=2π√ML2Eq

This has been linearised to simple harmonic motion (we have a linear restorative torque), and the minimum time τ for the arrangement to first go through the equilibrium point (ie where θ=0) is τ=T4=π2√ML2Eq