Multiple point-charges: A point charge Q = -500 nC and two unknown point charges, q₁ and q₂, are placed as shown in the figure. The electric field at the origin O, due to charges Q, q₁ and q₂, is equal to zero. The charge q₁ is closest to A) 130 nC. B) 76 nC. C) 150 nC. D) -76 nC. E) -130 nC.

Question

Quizplus · Accepted Answer

Since the electric field at the origin is zero, the net electric field due to the three charges is zero. Therefore, we can write:

kQ/r^2 + kq1/r1^2 + kq2/r2^2 = 0

where r is the distance from the origin to the charge Q, r1 is the distance from the origin to the charge q1, and r2 is the distance from the origin to the charge q2. Since we do not know the distances r1 and r2, we cannot solve this equation directly. However, we can use the fact that the charge q1 is closest to the origin to make an educated guess.

If we assume that q2 is farther away from the origin than q1, then we can write:

r2 > r1

Therefore, we can simplify the equation as follows:

kQ/r^2 + kq1/r1^2 > 0

This means that the electric field at the origin due to charges Q and q1 is not enough to cancel out the electric field due to charge q2. Therefore, charge q2 must be negative in order to produce an electric field that cancels out the electric field due to charges Q and q1.

Now, we can use the fact that the net charge of the system is zero to write:

Q + q1 + q2 = 0

Substituting q2 = -q1 - Q and rearranging, we get:

q1 = -Q/2

Therefore, we can calculate q2 as follows:

q2 = -q1 - Q = Q/2

Since q2 is positive, we know that our assumption above (that q2 is farther away from the origin than q1) is correct. Therefore, the charge closest to the origin is q1, which has a magnitude of:

|q1| = Q/2 = (-500 nC)/2 = -250 nC

However, the answer choices only give positive values for the charge. Therefore, we take the magnitude of q1, which is:

|q1| = 250 nC

This matches with Choice A, so our final answer is:

Answer: A

Multiple Point-Charges: a Point Charge Q = -500 NC and Two