Answer:
The third charge must be placed 0.548 m from q₁.
Explanation:
Let r = 3m be the distance between charge q₁ and q₂.
Let x be the distance between charge q₁ and charge q₃ (the third positive charge)
Then r - x is the distance between charge q₂ and q₃
Let the electrostatic force between q₁ and q₃ be F = kq₁q₃/x²
Let the electrostatic force between q₂ and q₃ be F' = kq₂q₃/(r - x)²
Since F + (-F') = 0 (the signs on the forces are different since the forces are in opposite directions)which is required when the net force on q₃ is zero, then
F - F' = 0
F = F'
So, kq₁q₃/x² = kq₂q₃/(r - x)²
q₁/x² = q₂/(r - x)²
[(r - x)/x]² = q₂/q₁
taking square-root of both sides,
(r - x)/x = ±√q₂/q₁
r/x - 1 = ±√q₂/q₁
r/x = 1 ±√q₂/q₁
x = r/(1 ±√q₂/q₁)
substituting the values of the variables r = 3 m, q₁ = 0.50 nC and q₂ = 10 nC
x = 3 m/(1 ±√10 nC/0.5 nC)
x = 3 m/(1 ±√20)
x = 3 m/(1 ± 4.472)
x = 3 m/5.472 or 3 m/-3.472
x = 0.548 m or -0.864 m
So the third charge must be placed 0.548 m to the right of q₁ or 0.864 m to the left of q₁.
Since we are concerned about the line of charge that connects q₁ and q₂, the third charge must be placed 0.548 m from q₁.
