An insulating sphere of mass m and positive charge q is attached to a spring with length h and spring constant ks and is at equilibrium as shown below:

An infinitely long wire with positive linear charge density λ is placed a distance l away from the charged mass at equilibrium as shown below (note that the position of the top of the spring is fixed):

The previous length of the spring was h. What is the new length of the spring in terms of h, q, ke (type "ke"), λ (type "lambda"), l, and ks (type "ks") as needed. Indicate multiplication with a "*" sign and division with a "/" sign. HINT: You can do this without considering the mass or gravitational force.

length of the spring =

h+ke*(lambda*q)/(l*ks)

Hai is check incorrect can u plz check if u write it correct and thx !!!

To find the new length of the spring, we need to consider the electric force between the charged sphere and the infinitely long wire.

First, let's calculate the electric force between the charged sphere and a small element of the wire. The force between two point charges q1 and q2 separated by a distance r is given by Coulomb's law:

F = (ke * q1 * q2) / r^2

In this case, the charge of the sphere is q, and the charge of the small element of the wire is dq. The distance between them is the perpendicular distance from the sphere to the wire, which is l. Therefore, the force dF between the sphere and this small element of the wire is:

dF = (ke * q * dq) / l^2

The total force F on the sphere due to the entire wire is obtained by integrating the force dF over the entire wire:

F = integral of [(ke * q * dq) / l^2]

To calculate the new length of the spring, we need to equate this electric force with the spring force, using Hooke's law:

F = ks * (h - new length)

where new length is the new length of the spring.

Now, let's solve for the new length of the spring:

(ke * q * dq) / l^2 = ks * (h - new length)

We can rearrange this equation to isolate the new length:

new length = h - [(ke * q * dq) / (ks * l^2)]

This equation represents the new length of the spring in terms of h, q, ke, λ, l, and ks.