Equilibrium of electric forces on charges on one line & formulas

In this post, we will take a few cases where the Equilibrium of electric forces on charges on one line happens. In every case, there are 3 charges on one line. We will list down the specific equilibrium condition of each such case and the formula of equilibrium for that scenario.

Scenario 1 (Equilibrium of electric forces)

As shown in the figure, all 3 charges are placed on one line.

Now, q₁ and q₂ are like charges while Q is the unlike charge. (This means, the charge in the middle is opposite in polarity to the other 2 charges)

1 ) When the force on q₁ is zero then

q₂/Q = (r₁ + r₂)²/r₁²

2 ) When the force on q₂ is zero then

q₁/Q = (r₁ + r₂)²/r₂²

3 ) When the force on Q is zero then

q₂/q₁ = r₂²/r₁²

Scenario 2 (Equilibrium of electric forces)

If all three charges are like charges then the condition for the equilibrium of Q is,

q₁/q₂ = r₁²/r₂²