In order to determine the position at which a third charge (2q) should be placed from charges q and -3q on the x-axis such that it experiences no force, we need to analyze the forces acting on the third charge and find the equilibrium condition.


Let's consider the charges q and -3q to be located at points A and B on the x-axis, respectively. We need to find the position of the third charge (2q) such that the net force acting on it is zero. This can be achieved when the electric forces exerted by charges q and -3q on the third charge are equal in magnitude and opposite in direction.


To achieve equilibrium, the electric forces exerted by charges q and -3q on the third charge must cancel each other out. This can be expressed mathematically as:

F1 = F2

where F1 is the force exerted by charge q on the third charge and F2 is the force exerted by charge -3q on the third charge.


The electric force between two charges can be calculated using Coulomb's law:

F = k * (|q1| * |q2|) / r^2

where F is the electric force, k is the electrostatic constant, q1 and q2 are the magnitudes of the charges, and r is the distance between the charges.

Let's assume that the third charge (2q) is located at point C on the x-axis, at a distance x from charge q (point A). The distance between charge -3q (point B) and charge 2q (point C) would be d - x.

Now, we can calculate the forces exerted by charges q and -3q on the third charge:

F1 = k * (|q| * |2q|) / x^2
F2 = k * (|-3q| * |2q|) / (d - x)^2


To achieve equilibrium, we set F1 equal to F2:

k * (|q| * |2q|) / x^2 = k * (|-3q| * |2q|) / (d - x)^2

Simplifying the equation, we get:

(q * 2q) / x^2 = (-3q * 2q) / (d - x)^2

2q^2 / x^2 = -6q^2 / (d - x)^2

Simplifying further, we have:

x^2 = (d - x)^2 / 3

Solving this quadratic equation, we can find the value of x that satisfies the equilibrium condition.


By solving the equation x^2 = (d - x)^2 / 3, we can determine the position at which the third charge (2q) should be placed from charges q and -3q on the x-axis such that it experiences no force.