Two particles having charges of 0.50~\text{nC}0.50 nC (q_1q 1 ) and 10~\text{nC}10 nC (q_2q 2 ) are separated by a distance of 3~\text{m}3 m (r = 3~\text{m}r=3 m). At what distance away from charge q1 along the line connecting the two charges can a third positive charge be placed such that it experiences a net electric force equal to zero?

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.

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