Multiple charges at different locations are said to be in equilibrium if the for

Question

Multiple charges at different locations are said to be in equilibrium if the force acting on any one of them is identical in magnitude and direction to the force acting on any of the others. Suppose we have two negative charges, one located at the origin and carrying charge -9e, and the other located on the positive x-axis at a distance d from the first one and carrying charge -36e. Determine the location, polarity and magnitude of a third charge whose placement would bring the entire system into equilibrium.

Explanation / Answer

Q1 = -9e @ X = 0
Q2 = -362 @ X = d
Let us assume third charge be Q3 at Distance x From Q1.
Distance of charge Q3 from Q2 = d-x

F1 = Force on Q1
F2 = Force on Q2
F3 = Force on Q3

At Equilibrium , F1 = F2 = F3

The two original charges are both negative, which mean they would repel each other. The third charge has to be positive and has to lie somewhere between them in order to counteract their repulsion force

Now Calculating Forces due to multi point charge -

F1 = - k* 9e*36e / d^2 + k*Q3*9e/x^2
F2 = + k* 9e*36e / d^2 - k*Q3*36e/(d-x)^2
F3 =  - k*Q3*36e/(d-x)^2 + k*Q3*9e/x^2

F1+F2+F3 = - 2k*Q3*36e/(d-x)^2 +  2k*Q3*9e/x^2 = 0
Q3*9 /x^2 = Q3 * 36 / (d-x)^2

Solving for x and Q3 -

Q3 = + 4e
x = d/3

Polarity = +ve
Location = d/3 from charge origin
Magnitude = 4e

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