A charge of +2q is fixed to one corner of a square, while a charge of -3q is fixed to the diagonally opposite corner. Expressed in terms of q, what charge should be fixed to the center of the square, so the potential is zero at each of the two empty corners?
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Physics - bobpursley, Sunday, September 2, 2007 at 11:10am
Potential is a scalar, so you can just add all four contributions.
Vt= -3q/r + 2q/r + 2X/r=0
solve for charge X
when i use the formula i can up with +1q and the system is saying that the answer is incorrect.
The equation you should be solving is
2q/a -3q/a + (sqrt 2)x/a = 0
Where a is the length of a side of the square. The corners with charges are a distance "a" away from the empty corners, but the center charge x is closer.
Multiplying bith sides by a gives
-q + sqrt 2 * x = 0
x = 0.707 q
Can you please explain how you get to x=0.707q.
I got how you got up to -q+sgrt2*X=0 but i keep coming up with a different answer than 0.707q.
solve the equation on paper to make it clear. you will end up doing -1/-(sqrt 2)
To find the charge that should be fixed to the center of the square in order for the potential to be zero at each of the two empty corners, we need to analyze the contributions from each charge.
Let q1 be the charge fixed to one corner, q2 be the charge fixed to the diagonally opposite corner, and q3 be the charge fixed to the center of the square. The potential at each empty corner can be calculated as follows:
Potential at empty corner 1: V1 = k * (q1 / r1)
Potential at empty corner 2: V2 = k * (q2 / r2)
where k is the electrostatic constant, r1 is the distance between empty corner 1 and the charge at the center of the square, and r2 is the distance between empty corner 2 and the charge at the center of the square.
Since we want the potential to be zero at both empty corners, V1 and V2 must both equal zero:
0 = k * (q1 / r1)
0 = k * (q2 / r2)
From the given information, we know that q1 = +2q and q2 = -3q. Plugging these values into the equations above, we get:
0 = k * (2q / r1)
0 = k * (-3q / r2)
Simplifying these equations, we find:
0 = 2kq / r1
0 = -3kq / r2
Now, since the potential is a scalar quantity, we can add up the potentials at each empty corner to get the total potential:
Total potential = V1 + V2
Since V1 = 0 and V2 = 0, the total potential is also zero:
0 = V1 + V2 = 0 + 0 = 0
Therefore, the potential at each empty corner is zero. This means that the sum of the contributions from each charge must be equal to zero. We can express this as:
0 = k * (q1 / r1) + k * (q2 / r2) + k * (q3 / r3)
Substituting q1 = +2q and q2 = -3q, we have:
0 = k * (2q / r1) + k * (-3q / r2) + k * (q3 / r3)
Now, we need to solve this equation for q3, the charge fixed to the center of the square.
0 = 2kq / r1 - 3kq / r2 + kq3 / r3
Multiplying through by r1 * r2 * r3 to eliminate the denominators, we get:
0 = 2kq * (r2 * r3) - 3kq * (r1 * r3) + kq3 * (r1 * r2)
Now, rearranging the equation, we have:
2k * (r2 * r3) - 3k * (r1 * r3) = -kq3 * (r1 * r2)
Dividing through by -k * (r1 * r2) to isolate q3, we find:
q3 = (2 * r2 * r3 - 3 * r1 * r3) / (r1 * r2)
Therefore, the charge that should be fixed to the center of the square, in terms of q, is:
q3 = (2 * r2 * r3 - 3 * r1 * r3) / (r1 * r2)
You can now substitute the values for r1, r2, and r3 into the equation to find the numerical value of q3.