Three identical conducting spheres are located at the vertices of an equilateral triangle ABC. Initially the charge the charge of the sphere at point A is q A =0 and the spheres at B and C carry the same charge q A =q B =q. It is known that the sphere B exerts an electrostatic force on C which has a magnitude F=4N. Suppose an engineer connects a very thin conducting wire between spheres A and B. Then she removes the wire and connects it between spheres A and C. After these operations, what is the magnitude of the force of interaction in Newtons between B and C?
F(BC) =kq²/a².
q´(B)=q´(A) =(q+0)/2=q/2,
q´´(A) =q´(C) = {q´(A)+c(C)}/2 = 3q/4,
F´(BC) =k q´(B)• q´(C)/a² =
=(3/8)• kq²/ a²=3•4/8=3.2 = 1.5 N
To find the magnitude of the force of interaction between B and C after the wire is moved, we need to understand how the charges redistribute.
When the thin conducting wire is connected between spheres A and B, the charges redistribute to reach equilibrium. Since sphere B initially had a charge of q, it will now share half its charge with sphere A, resulting in each sphere having a charge of q/2.
Next, when the wire is moved and connected between spheres A and C, the charges redistribute again to reach a new equilibrium. Now, the charge on sphere B (q/2) is shared equally with sphere C, resulting in both spheres having a charge of q/2.
To calculate the magnitude of the force of interaction between B and C, we'll use Coulomb's law:
F = k * (|q1| * |q2|) / r^2
Where F is the electrostatic force, q1 and q2 are the charges on the spheres (|q1| = q/2, |q2| = q/2), r is the distance between the spheres, and k is the electrostatic constant.
In the initial setup, F = 4N. Let's assume the distance between the spheres is d.
So, using Coulomb's law for the initial setup:
4N = k * (|q1| * |q2|) / d^2
Now, let's calculate the magnitude of the force of interaction between B and C after the wire is moved:
F' = k * (|q1| * |q2|) / d'^2
Since the charges on B and C after the wire is moved are both q/2, we have:
F' = k * [(q/2) * (q/2)] / d'^2
Now, to find the relationship between d and d', we need to consider the equilateral triangle formed by the spheres. The distance between any two spheres is equal to the length of a side of the triangle, which we'll call L.
Using trigonometry, we can determine that d = L * √(3).
After the wire is moved, the distance between A and C remains the same as d, as the original equilateral triangle remains unchanged.
Therefore, d' = d = L * √(3).
Substituting d = L * √(3) into F', we get:
F' = k * [(q/2) * (q/2)] / (L * √(3))^2
F' = k * (q^2 / 4) / (3L^2)
Now, to find the ratio of F' to F, we can divide them:
F' / F = [k * (q^2 / 4) / (3L^2)] / [k * (q^2 / 4) / d^2]
F' / F = (d^2 / (3L^2))
Since d = L * √(3), we can substitute it:
F' / F = [(L * √(3))^2 / (3L^2)]
F' / F = (3L^2 / (3L^2))
F' / F = 1
Therefore, the magnitude of the force of interaction between B and C remains the same after the wire is moved, and it is still 4N.