Answer to Question #108861 in Electric Circuits for Uday

Question #108861
Two point charges +4q and +q are separated by distance r, where should a third point charge Q be placed so that the whole system remains in equilibrium?
1
Expert's answer
2020-04-13T09:58:10-0400

Let place the charge Q on the line between +4q and q charges as shown on the figure.





Then, according to the Coulomb's low, the interaction force between +4q and Q charges is:

F1=k4qQx2.F_1 = k\dfrac{4q\cdot Q}{x^2}.\\

The interaction force between +q and Q charges is:

F2=kqQ(rx)2,F_2 = k\dfrac{q\cdot Q}{(r - x)^2}, \\ where rr is the distance between +4q and +q and kk is the Coulomb's constant.

In the equilibrium state both forces are equal, thus obtain the equation:

k4qQx2=kqQ(rx)2k\dfrac{4q\cdot Q}{x^2} = k\dfrac{q\cdot Q}{(r - x)^2} .

Solving it for xx , obtain:

4x2=1(rx)24(rx)2=x22(rx)=xx=23r.\dfrac{4}{x^2} = \dfrac{1}{(r - x)^2} \Rightarrow 4(r-x)^2 = x^2 \Rightarrow 2(r-x) = x \\ x = \dfrac{2}{3}r.\\


Answer. The charge Q should be placed at the distance 2/3r from the charge +4q.


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