Question

A particle with charge q on the negative x axis and a second particle with charge 2q on the positive x axis are each a distance d from the origin. Where should a third particle with charge 3q be placed so that the magnitude of the electric field at the origin is zero? (Use any variable or symbol stated above as necessary.)

Answer 1

The charge 3q must be placed at a distance of 1.73d away from the origin.

Step-by-step explanation:

The electric field at the origin due to q at distance, d from the origin is E = q/4πε₀d².

The electric field at the origin due to 2q at distance, d from the origin is E' = -2q/4πε₀d². (It is negative since it its directed towards the negative x-axis)

Now, the net electric field at the origin due to both charges is thus E" = E + E' = q/4πε₀d² + (-2q/4πε₀d²) = q/4πε₀d² - 2q/4πε₀d² = -q/4πε₀d².

The electric field due to 3q at a distance x from the origin is E₁ = 3q/4πε₀x².

Now, the net electric field E' must cancel out E₁ so that the electric field at the origin is zero.

So, E' + E₁ = 0

E' = -E₁

-q/4πε₀d² = -3q/4πε₀x²

1/d² = 3/x²

taking inverse of both sides, we have

x²/3 = d²

multiplying both sides by 3, we have

x² = 3d²

taking square root of both sides, we have

x = (√3)d

x = ±1.73d

So, the charge 3q must be placed at a distance of 1.73d away from the origin.