Three point charges are arranged along the x axis. Charge q1 = -4.10 nC is located at x= 0.250 mand charge q2 = 2.20 nC is at x= -0.320 m . A positive point charge q3 is located at the origin. What must the value of q3 be for the net force on this point charge to have magnitude 3.10 μN ? (ans in nC)

What is the direction of the net force on q3? + or - direction

Where along the x axis can q3 be placed and the net force on it be zero, other than the trivial answers of x=+[infinity] and x=−[infinity]? (ans in m)

1) q₃ is approximately -7.58 nC

2) The direction of the net force on 'q₃' is the negative '-' direction

3) For the net force on 'q₃' to be zero, 'q₃' can be placed at x = -0.079 m or x = -1.881 m

Explanation:

1) The force, 'F', between two charged spheres is given as follows;

The net force acting on the point charge, 'q₃', is given as follows;

By substituting the given values, we have;

K = 9 × 10⁹ N·m²·C⁻²

q₃ ≈ -7.58 nC

2) Given that the negative charge, 'q₁' (-4.10 nC), is located at x = 0.250 m, which is on the positive, '+' side of the origin, it will repel the negatively charged 'q₃', to the '-' direction. q₃ will also be attracted to the '-' direction by the positively charged 'q₂' which is at -0.320 m on the negative side of the origin

The net force's direction on q₃ will be in the '-' direction

3) For zero net force, we have;

The distance between the given point charges, r = 0.250 - (-0.320)) = 0.57 m

Let 'r1' represent the distance between 'q1' and 'q3', therefore, the distance between 'q2' and 'q3' is 0.57 - r1

By substitution, we have;

From which we have;

(r₁)²×4.10 = 2.20 × (0.57 - r₁)²

From the above equation, we have;

95,000·r₁²+ 125,400·r₁-35739 = 0

Solving, using a graphing calculator we get;

r₁ ≈ 0.241 or r₁ ≈ -1.561

Where, 'r₁', is measured from 'q₂', therefore, we have;

r₁ = 0.241 + (-0.320) ≈ -0.079 or r₁ = -1.561 + (-0.320) ≈ -1.881

Therefore, the charge 'q₃' can be placed at x = -0.079 or x = -1.881 for the ne force on it to be zero