Two positive point charges are placed on the x-axis. One, of magnitude 4Q, is placed at the origin. The other, of magnitude Q is placed at x=3 m. Neither charge is able to move. Where on the x-axis in meters can I place a third positive point charge such that the magnitude of the net force on the third charge is zero?
Lets assume a charge of q is placed at a point (x,o) where 0<x<3.
Your net force on that middle charge will be the sum of two electrostatic forces. Your plan is to use a distance of x for the force calculation between the leftmost charges and a distance of 3-x for the force calculation between the rightmost charges. These two forces should be equal to achieve equilibrium.
Basically, thanks to Coulomb's Law, I have 4Q/(x2) = Q/((3-x)2) after simplification. This yields
x^2-8x+12=0
(x-2)(x-6)=0
x=2 or x=6
Now just note that x has to be between 0 and 3 meters because that's the only way the field directions will oppose each other.
Anyways, this is quite a nice application of Coulomb's law. However, next time please refrain from posting live brilliant problems.
To determine the position on the x-axis where a third positive point charge can be placed such that the magnitude of the net force on it is zero, we need to consider the electrostatic forces exerted by the existing charges.
Let's break down the problem step by step:
Step 1: Identify the charges and their magnitudes:
- Charge 1: Magnitude = 4Q (at the origin, x = 0)
- Charge 2: Magnitude = Q (at x = 3 m)
- Charge 3: Unknown magnitude (we need to find this)
Step 2: Determine the direction of the net force on Charge 3:
- The force between Charges 1 and 3 depends on their charges and the distance between them.
- The force between Charges 2 and 3 depends on their charges and the distance between them.
- Since both Charges 1 and 2 are positive, their forces will repel Charge 3.
Step 3: Calculate the force between Charge 1 and Charge 3:
- The electrostatic force between two point charges is given by Coulomb's Law:
F₁₃ = (k * |q₁ * q₃|) / r₁₃²
Where F₁₃ is the force between Charge 1 and Charge 3,
k is the electrostatic constant (approximated as 9 x 10^9 Nm²/C²),
q₁ is the magnitude of Charge 1,
q₃ is the magnitude of Charge 3, and
r₁₃ is the distance between Charge 1 and Charge 3.
Step 4: Calculate the force between Charge 2 and Charge 3:
- Similarly, apply Coulomb's Law to find the force between Charges 2 and 3:
F₂₃ = (k * |q₂ * q₃|) / r₂₃²
Where F₂₃ is the force between Charge 2 and Charge 3,
q₂ is the magnitude of Charge 2, and
r₂₃ is the distance between Charge 2 and Charge 3.
Step 5: Find the position where the net force is zero:
- For the net force on Charge 3 to be zero, the magnitudes of the forces F₁₃ and F₂₃ should be equal.
- Set F₁₃ = F₂₃ and solve for the unknown distance r₁₃.
- Once you find the distance r₁₃, subtract it from x = 3 m to determine the position on the x-axis where Charge 3 should be placed.
By following these steps, you should be able to determine the position on the x-axis where the third positive point charge should be placed.