Introduction:
In this scenario, we have a point charge q placed at a distance 2R from the center of an isolated conducting sphere of radius R. We need to find the minimum charge that needs to be placed on the sphere so that the force on the point charge will be zero.

Understanding the Problem:
To solve this problem, we need to consider the concept of electrostatic force and the principle of electric field inside a conductor.

Electrostatic Force:
The electrostatic force between two charged particles is given by Coulomb's Law:
F = k * (q1 * q2) / r^2
where F is the force between the charges, q1 and q2 are the magnitudes of the charges, r is the distance between the charges, and k is the Coulomb's constant.

Electric Field inside a Conductor:
Inside a conductor, the electric field is zero. This is because any excess charge on a conductor distributes itself in such a way that the electric field inside the conductor is zero.

Solution:
To find the minimum charge needed on the conducting sphere, we need to ensure that the net force on the point charge q is zero.

Step 1: Calculate the force between the point charge q and the conducting sphere.
Using Coulomb's Law, the force between q and the conducting sphere is given by:
F = k * (q * Q) / (2R)^2
where Q is the charge on the conducting sphere.

Step 2: Set the force equal to zero and solve for Q.
Since we want the net force on q to be zero, we set F = 0:
0 = k * (q * Q) / (2R)^2
Simplifying the equation, we get:
Q = -((2R)^2 * q) / k

Step 3: Calculate the minimum charge needed on the conducting sphere.
To find the minimum charge needed on the conducting sphere, we take the absolute value of Q:
Q = ((2R)^2 * q) / k

Step 4: Calculate the minimum charge needed on the conducting sphere in terms of q.
Substituting the value of k (Coulomb's constant) and simplifying the equation, we get:
Q = (4 * R^2 * q) / (4πε₀)
where ε₀ is the permittivity of free space.

Conclusion:
The minimum charge that needs to be placed on the conducting sphere in order to create a net force of zero on the point charge q is given by Q = (4 * R^2 * q) / (4πε₀).