A circular metal plate of mass m and area A is suspended from three vertical ins

Question

A circular metal plate of mass m and area A is suspended from three vertical insulating strings, such that its plane is horizontal. A second metal plate, identical to the first, hangs directly below and parallel to the first plate from a spring which is connected to the centre of each plate. This metal spring has spring constant K, natural length, l, and a negligible mass, and is connected to each plate by metal fixings. Any charge on the system is removed by temporarily earthing the plates, after which the separation of the plates is L_0, where L_0^2

Explanation / Answer

spring constant = k
mass = m
natural length = l
final plate seperation = lo
now from force balance
k(lo - l) = mg
lo = mg/k + l

also, 3*To = k(lo - l) + mg [ To is tension in the string]
so, To = 2mg/3

as there is electric field aplied, all the charge will migrate to lower plate, as the field is pointed from lower to upper plate
charge entering bottom plate = q
so new plate seperation = l'
then q^2/4*pi*e*(l')^2 = k(l' - l)

also, potential difference , V = E(l') = q/C, where C = Ae/l'
so, El' = qAe/l'
q = El'^2/Ae

so, E^2*l'^4 / A^2 e^2 * 4 *pi*e*i'^2 = k(l' - l)
E^2*l'^2/A^2*e^3*4*pi = kl' - kl
l' = [k + sqroot(k^2 - E^2/A^2*e^3*pi)]/2E^2/A^2*E^3*4*pi
lo - l' = mg/k + l - [k + sqroot(k^2 - E^2/A^2*e^3*pi)]/2E^2/A^2*E^3*4*pi

and T' - T = (mg + k(l' - l) - qE)/3 - 2mg/3 ={} k{[k + sqroot(k^2 - E^2/A^2*e^3*pi)]/2E^2/A^2*E^3*4*pi - l} - E{[k + sqroot(k^2 - E^2/A^2*e^3*pi)]/2E^2/A^2*E^3*4*pi}^2/Ae - mg}/3

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