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Two point charges in free space are Q1=10nC at (5,0,0) and Q2= 25nC at (-5,0,0). Where shouldQ3=20nC be located such that the force on Q3 is zero

Question

Two point charges in free space are Q1=10nC at (5,0,0) and Q2= 25nC at (-5,0,0). Where should Q3=20nC be located such that the force on Q3 is zero?

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Solution

The force on Q3 due to Q1 and Q2 must be equal and opposite for the net force to be zero.

Let's consider two possible positions for Q3: one along the line joining Q1 and Q2 to the right of Q1 (position A) and the other to the left of Q2 (position B).

  1. Position A (x,0,0) where x > 5:

The force on Q3 due to Q1 is given by Coulomb's law:

F31 = kQ1Q3/|r3 - r1|^2 = kQ1Q3/(x - 5)^2

Similarly, the force on Q3 due to Q2 is:

F32 = kQ2Q3/|r3 - r2|^2 = kQ2Q3/(x + 5)^2

For the net force on Q3 to be zero, F31 = F32. Therefore:

Q1/(x - 5)^2 = Q2/(x + 5)^2

Solving this equation will give the value of x for position A.

  1. Position B (-x,0,0) where x > 5:

Similarly, we can write:

Q1/(x + 5)^2 = Q2/(x - 5)^2

Solving this equation will give the value of x for position B.

Note: k is Coulomb's constant, r1, r2, and r3 are the positions of Q1, Q2, and Q3 respectively.

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