Question

idential point charges of 1.7e-6 C are fixed diagonally on opposite corners of a square. A third charge is then fixed at the center of the square, such that it causes the potentials at the empty corners to change signs without changing their magnitudes. Find the sign and magnitude of the third charge in the center of the square.

i am absolutely clueless.

Answer 1

Let the charges at the two opposite corners be q, and the added charge at the center be q'. Before q' is added, the potential at the empty corners is
2 k q/a, where a is the length of a side of the square.

The distance of the center of the square from either empty corner is a/sqrt2

The potential at the emptry corners with q' added must be

-2kq/a = 2kq/a + kq' sqrt2/a

The k's and a's cancel and you are left with
-2q = +q' *sqrt2
q' = -sqrt2*q (since 2/sqrt2 = sqrt2)