should this third charge be added if it will experience no net electrical force?

Question

Two small spheres, with charges 1.6*10^-5 C and 6.4*10^-5 C, are 2.0 m apart.

The charges have the same sign.

Where, relative to these two spheres, should a third

sphere, of opposite charge 3.0*10^-6 C, be placed if the

third sphere is to experience no net electrical force?

Do we really need to know the charge or sign of the third object?

Answer 1

Let's first try to solve this by assuming we don't need the actual charge and sign of the third object. I will call this charge Q where Q is an arbitrary, nonzero charge. Let q₁ = 1.6 * 10^-5 C at x₁ = 0.0 m and q₂ = 6.4 * 10^-5 C at x₂ = 2.0 m.

Our goal is to find the location where the third charge, Q, will have a net force of 0 N. This will either be to the left of q₁ (x_Q < 0), between q₁ and q₂ (0 < x_Q < 2), or to the right of q₂ (2 < x_Q). If the third charge is not on the line connecting the charges, we will have components of the force either pushing the charge away from the x-axis (if Q is positive) or pulling it towards the x-axis (if Q is negative) because q₁ and q₂ have the same sign of charge.

If x_Q is to the left of q₁, then the force from both charges will be in the same direction (positive x if Q is negative, negative x if Q is positive). If x_Q is to the right of q₂, then the force from both charges will still be in the same direction (negative x if Q is negative, positive x if Q is positive).

Without knowing anything about Q, we have determined that it has to lie on the x-axis between x = 0 m and x = 2 m. Now we can use Coulomb's law to determine the net force on Q. Since I can't use unit vectors on here to show direction, we will assume positive is in the positive x direction and negative is in the negative x direction.

(1) F_net = F₁ + F₂

(2) F₁ = k * (q₁ * Q) / (x_Q)²

(3) F₂ = - k * (q₂ * Q) / (2 - x_Q)²

Here, the sign of the force will be taken care of by the sign of Q. If Q is positive, then F₁ points away from q₁ and F₂ points away from q₂. Similarly, if Q is negative, F₁ points towards q₁ and F₂ points away from q₂.

Now we will plug (2) and (3) into (1) and set F_net = 0 and simplify.

(4) 0 = k * (q₁ * Q) / (x_Q)² - k * (q₂ * Q) / (2 - x_Q)²

(5) (k * Q) * (q₁ / (x_Q)²) = (k * Q) * (q₂ / (2 - x_Q)²)

Here we can see both sides have a common factor of k * Q which cancel out. The remaining equation does not depend on Q so we do not need to know the charge or sign of Q. The only requirement is that Q is nonzero. If Q was zero, then you could place it anywhere and the net force would be 0.

To get the final result, we just need to solve (5) for x_Q.

(6) sqrt(q₂ / q₁) = (2 - x_Q) / x_Q

(7) x_Q * sqrt(q₂ / q₁) = 2 - x_Q

(8) x_Q * (1 + sqrt(q₂ / q₁)) = 2

(9) x_Q = 2 / (1 + sqrt(q₂ / q₁))

Plugging q₁ and q₂ into (9), we find that x_Q = 0.67 m.

As you said in your question, we do not actually need to know the charge or sign of the third object for this problem because it cancels out with the net force = 0. If we were looking at a point where there was an electrical force, then we would need to know Q to solve this problem.