An insulating sphere of mass m and positive charge q is attached to a spring with length h and spring constant k_s and is at equilibrium. An infinitely long wire with positive linear charge density lamda 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 , K_e (type "ke"), lambda

h-(2*ke*q*lambda)/(ks*l)

To determine the new length of the spring, we need to consider the electrostatic force between the charged sphere and the wire.

The electrostatic force between a charged sphere and an infinitely long wire can be calculated using Coulomb's Law. Coulomb's Law states that the electrostatic force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

In this case, the force experienced by the charged sphere due to the wire can be calculated as:

F = (k_e * q * λ) / l

Where:
F is the force,
k_e is the electrostatic constant (usually denoted as ke or k_e),
q is the charge of the sphere,
λ is the linear charge density of the wire, and
l is the distance between the sphere and the wire.

Since the sphere is attached to a spring, any force acting on the sphere will cause the spring to stretch or compress. Hooke's Law can be used to relate the force exerted on the spring to the change in length of the spring.

Hooke's law states that the force exerted by a spring is directly proportional to the displacement produced by the force. Mathematically, it can be expressed as:

F = k_s * Δl

Where:
F is the force exerted by the spring,
k_s is the spring constant, and
Δl is the change in length of the spring.

Since the sphere is in equilibrium, the force exerted by the wire is equal in magnitude and opposite in direction to the force exerted by the spring. Therefore, we can equate these two forces:

(k_e * q * λ) / l = k_s * Δl

To find the new length of the spring, we isolate Δl:

Δl = (k_e * q * λ * l) / (k_s)

Finally, we can calculate the new length of the spring in terms of h, q, k_e, and λ by adding this change in length to the initial length of the spring:

New length of the spring = h + Δl

Thank you very much