Three charged balls are placed so that each is at a different corner of a square

Question

Three charged balls are placed so that each is at a different corner of a square, as shown in the figure below. Balls 1 and 3 each have positive charges, but the sign of the charge on ball 2 is not shown. The figure also shows the net force acting on each of the balls—the only forces that matter here are those associated with the interactions between the charges.

(a) Based on the information given in the figure, select all the true statements from the following list.

Ball 2 has a positive charge.

Ball 2 has a negative charge.

There is not enough information to determine the sign of the charge on ball 2.

The charge on ball 1 is larger than the charge on ball 3.

The charge on ball 1 is smaller than the charge on ball 3.

The charge on ball 1 is equal to the charge on ball 3.

(b) Rank the balls, from largest to smallest, based on the magnitude of the charge on them. (Use only ">" or "=" symbols. Do not include any parentheses around the letters or symbols.)

(c) Determine the ratio of the magnitude of the charge on ball 1 to the magnitude of the charge on ball 2.

Please help! and show work! i'm extra confused on part c :( the answer for part c is not 1/4 or 4

Explanation / Answer

a) assume right side as +ve x axis and up as +ve y axis.
so force on 3 due to 1.

F13 = (kq1q3) / (L^2 + L^2) [ cos45i - sin45j ]

F13 = kq1q3/ 2sqrt(2)L^2 (i - j)

and force on 3 due to 1.

F23 = kq2q3 / L^2 (i)

and this should be (-i) so that means q2 is negative.

and kq2q3 / L^2 = kq1q3/ 2sqrt(2)L^2

q1 = 2sqrt(2)q2

so q1 > q2
Ball 2 has a negative charge.

------------

force on q2 is coming towards 3.

that means F23 is greater than F12.

that means q3 is greater q1.

The charge on ball 1 is smaller than the charge on ball 3.

-------------------

b) q3 > q1

and q1 > q2

q3 > q1 > q2

c) q1 = 2sqrt(2)q2

q1/q2 = 2 sqrt(2) = 2.83

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