How Does Charge Interaction Affect Spring Dynamics?

[Lunar_Lander]

Homework Statement

Charge Q₁ is fixed to the ceiling by means of a spring with the spring constant D and is moved from its rest position (which is at y=h) by the two charges Q₂. The gravitational force may be neglected.

a) Determine the resulting force on Q₁ in dependency of y.

b) The distance of the two charges Q₂ should now be zero (a=0). Determine the charge Q₂ as a function of y so that the resulting force on the charge Q₁ becomes zero.

Homework Equations

Field equations of radial electric fields, force of a spring.

The Attempt at a Solution

I do not really have an idea for a), for b) however I tried to equalize the electric force and the spring force and then I solved for Q₂. That looked like this:
$Q_2=-\frac{D*4*\pi*\epsilon_0*y^3}{2*Q_1}$. Can this be right?

 

[anathema]

For part a), the resulting force on Q1 can be found by considering the forces acting on it. The spring force is given by F = -k(y-h), where k is the spring constant and y is the displacement from the rest position. The electric force due to Q2 is given by F = kQ1Q2/y^2, where Q1 and Q2 are the two charges and y is the distance between them.

Therefore, the resulting force on Q1 is F = -k(y-h) + kQ1Q2/y^2. This can be further simplified by factoring out the spring constant k, giving F = k(-y+h+Q1Q2/y^2).

For part b), the resulting force on Q1 should be zero when the distance between Q1 and Q2 is zero. This means that the electric force should be equal and opposite to the spring force. Setting these forces equal to each other and solving for Q2 gives Q2 = -k(y-h)/Q1. This is the charge that would result in a zero net force on Q1 when the distance between Q1 and Q2 is zero.

Hope this helps! Let me know if you have any further questions or need clarification. Keep up the good work in your studies.