What should be the value of Q?

I am doing the basics of coulomb's law,can anyone explain this to me?

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# [Physics] Electrostatics Equilibrium problem

coulombs-lawelectrostaticshomework-and-exercises

What should be the value of Q?

I am doing the basics of coulomb's law,can anyone explain this to me?

## Best Answer

By symmetry, you need only consider the force on one of the charges on the corner, say the top right, due to the other four. There will be a repulsion from the other corners pushing it away from the centre, so you must pick Q to balance this with an attractive force of just the right size.

One way to work this out is to calculate the vector forces (proportional to $\frac{\vec{r}_1-\vec{r}_2}{|\vec{r}_1-\vec{r}_2|^3}$) due to each of the 4 other charges, and add them to get the total force. You should get some coefficient times the diagonally pointing direction vector, and the right choice of $Q$ will set this coefficient to zero.

An alternative way, if you've met the electrostatic potential, would be to specify that the top-right charge is at a stationary point of the potential due to the other four. To do this, allow the charge to sit at some other point on the diagonal, $t\vec{i}+t\vec{j}$ (other directions don't need to be considered by symmetry), and compute the potential as a function of $t$. You can fix Q by specifying that $t=a$ is a stationary point of this potential.

To check you got your working right, the answer in the end is $Q=-(\frac14+\frac1{\sqrt{2}})q$.