Three point charges are fixed to the corners of a square, one to acorner, in suc

Question

Three point charges are fixed to the corners of a square, one to acorner, in such a way that the net electric field at the emptycorner is zero. Do these charges all have(a) the same sign?
(b) the same magnitude ? - Which of the following are true?

1. Since the field at the empty corner due to the charge placed atthe diagonally opposite corner will have equal x and y components,the charges placed at the other two corners must have the samemagnitude.

2. It is certainly possible for all three occupied corners of thesquare to have charges with the same magnitude and still produce azero electric field at the unoccupied corner of the square.

3. It is impossible to have any arrangement of charges, with anycombination of sign and magnitude, at 3 of the 4 corners of asquare such that the electric field at the unoccupied corner iszero.

4. The charge placed at the diagonally opposite corner to the emptycorner must have a charge opposite in sign to the identical chargesplaced at the other occupied corners; and have a charge whosemagnitude is larger than that of the other two charges.

5. The charges placed at thetwo corners along the edge of the square away from the empty musthave the same sign. In that way the net field at the empty cornerdue to the two charges alone will lie along the diagonal of thesquare.

6. It is certainly possible for all three charges at the occupiedcorners of the square to have the same sign and still produce zerofield at the unoccupied corner of the square.

Explanation / Answer

The sign of the charges must be taken intoaccount in finding the electric field.
Here's a hint: Somehow the fields from each of the three chargesmust "cancel out" at the fourth corner. So, what does that tell youabout the signs of the charges?
Note that what counts is the distance from the charges to thefourth corner. Note that q1 and q3 are the same distance away fromthe empty corner, but q2 is on a diagonal and thus furtheraway.

Since q1 and q3 are on opposite sides of the diagonal, what can yousay about their charges? Are they the same? Different?

Draw yourself a picture showing the fields produced by the threecharges at that fourth corner. Remember that the field from apositive charge points away from the charge. The sign of the charges must be taken intoaccount in finding the electric field.
Here's a hint: Somehow the fields from each of the three chargesmust "cancel out" at the fourth corner. So, what does that tell youabout the signs of the charges?
Note that what counts is the distance from the charges to thefourth corner. Note that q1 and q3 are the same distance away fromthe empty corner, but q2 is on a diagonal and thus furtheraway.

Since q1 and q3 are on opposite sides of the diagonal, what can yousay about their charges? Are they the same? Different?

Draw yourself a picture showing the fields produced by the threecharges at that fourth corner. Remember that the field from apositive charge points away from the charge.
To make things easier to describe, let's imagine the empty corneris at the origin and the opposite diagonal (where q2 is) issomewhere on the y-axis. Thus q1 (on the left side of the y-axis)and q3 (on the right side) have the same y-coordinate, but oppositex-coordinates. Got it?
If both charges on opposite sides of the diagonal were positive youwould have one charge pointing in the -x, -y direction and onepointing in the +x, +y direction.
Careful. Using my coordinate system the field from q1 will havecomponents along the +x and -y directions; the field from q3 willhave components along the -x and -y directions. So if q1 and q3 areboth positive and equal, then the x-components would cancel, butnot the y-components--those will add.
So those two would cancel each other out.
The net field from q1 and q3 will point in the -y direction.
The other charge (on the diagonal) is a length = sq.root(a^2+b^2)away from the empty corner, according to pythagorean's theorem. Butsince it is a square, all of the sides are equal length of d so aand b would be equal. Thus the above distance equation would =sq.root(d^4). So how would a negative charge at this corner allowthe empty corner to have an electric field of 0?
If the sides are length d, the diagonal is length sqrt{2} d.Since the field from q1 and q3 points down (-y direction), tocancel that field you need to add a field pointing up (+ydirection). A negative q2 will do that, but it needs to be theright size charge to cancel the other fields exactly.

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