Irodov Solutions : Problem 3.4

Two positive charges q₁ and q₂ are located at the points with position vectors r₁ and r₂ respectively. Find a negative charge q₃ and a position vector r₃ of the point at which it has to be placed for the forces acting on each of the three charges to be equal to zero.

Firstly, the charge q₃ should be placed at a point where the resultant electrostatic field due to the other two charges q₁ and q₂ is zero. Secondly, the magnitude of the charge q₃ should be so chosen that it can cause the other charges to come in the state of equilibrium.

Let P be the point at which the charge q₃ should be placed. Since we want q₃ to be in equilibrium, the point P must be a null point i.e. the net field at point P must be zero. Clearly, the point P will lie on the line connecting the two charges q₁ and q₂ as shown below. 
For the equilibrium of charge q₃ we can write, 

Thus we get the the required position vector r₃ and the value of charge q₃ for the whole system of charges to stay in electrostatic equilibrium. 


The three point charges are staying in equilibrium under their electrostatic interactions only. Therefore by Earnshaw's theorem, the equilibrium established is unstable in nature. Even an infinitesimal displacement given to any of the three charges would be sufficient to destroy the equilibrium of the electrostatic system. 

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