Two fixed, non-conductive, isolated rings of radii 25 cm are electrified and brought close to each other so that they are parallel. The distance between them is 1 m. You know that one of them has a charge of 1 nC(Nano Coulomb) evenly distributed on it, but you are not sure about the other. In order to find out what's the charge on the other, you take a charged ball and move it across the axis of the rings until you find its equilibrium position. You measure where that is and get that the equilibrium position is 25 cm from the ring with the known charge. What's the charge in nC of the second ring?
Details and assumptions:
The second ring also has evenly distributed positive charge on it.
I solved it like this
let charge of the ball be Q
let charge of unkown ring be Q2
let the place where there is equilbrium be
P. This point P is 0.25m from the ring with known charge and and hence 0.75m from the ring with unknown charge.
Now applying coulomb's law at this point P from both the rings and this force must be equal as the point is in chrge equilibrim.
So at point p
(k*Q2*Q)/(0.75)^2 = (k*1*10^-9*Q)/(0.25)^2
[As known charge is 1nC)
Solving we get Q2 = 9 nC.
But the answer is incorrect so what should be the correct answer ?
PLEASE HELP !
Electric field on the axis of the ring at the distance ‘x’ from the center of it is
E = k xq/{sqrt(R²+x²)}³
k=1/4πε₀
The test charge is in equilibrium =>
E₁=E₂
k x₁q₁/{sqrt(R²+x₁²)}³=
=k x₂q₂/{sqrt(R²+x₂²)}³,
q= x₁q₁{sqrt(R²+x₂²)}³/{sqrt(R²+x₁²)}³=
=0.25•1•10⁻⁹•{sqrt(0.25²+0.75²)}³/{sqrt(0.25²+0.25²)}³=
=2.8•10⁻⁹ C = 2.9 nC