Consider three charged particles placed along the x-axis. The particle with q1 = 9.30 nC is at x = 6.87 m and q2 = 4.09 nC is at x = 0.600 m. Where must a positive charge, q3, be placed such that the resultant force on it is zero?
x=?m
To find the position, x, where a positive charge, q3, must be placed such that the resultant force on it is zero, we can use the principle of superposition of forces.
The force between two charged particles is given by Coulomb's Law:
F = (k * |q1| * |q2|) / r^2
Where:
F is the force between the charges,
k is the electrostatic constant (k = 8.99 × 10^9 N m^2/C^2),
|q1| and |q2| are the magnitudes of the charges,
and r is the distance between the charges.
Since we want the resultant force on q3 to be zero, the forces exerted on q3 by q1 and q2 must cancel each other out.
Let's calculate the individual forces, F1 and F2, exerted on q3 by q1 and q2:
F1 = (k * |q1| * |q3|) / (x - x1)^2
F2 = (k * |q2| * |q3|) / (x - x2)^2
Where:
x1 and x2 are the positions of q1 and q2, respectively.
For the resultant force to be zero, the magnitudes of F1 and F2 must be equal:
F1 = F2
Setting the equations for F1 and F2 equal to each other, we get:
(k * |q1| * |q3|) / (x - x1)^2 = (k * |q2| * |q3|) / (x - x2)^2
Now, we can simplify the equation by cancelling out |q3| and k on both sides:
|q1| / (x - x1)^2 = |q2| / (x - x2)^2
Next, we can cross multiply to get rid of the fractions:
|q1| * (x - x2)^2 = |q2| * (x - x1)^2
Expanding and simplifying:
|q1| * (x^2 - 2x*x2 + x2^2) = |q2| * (x^2 - 2x*x1 + x1^2)
Now, we can gather the terms:
(|q1| - |q2|) * x^2 + 2(|q2| * x1 - |q1| * x2) * x + (|q1| * x2^2 - |q2| * x1^2) = 0
This is a quadratic equation in terms of x. We can use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / 2a
Where:
a = |q1| - |q2|,
b = 2(|q2| * x1 - |q1| * x2),
c = |q1| * x2^2 - |q2| * x1^2.
Now, substitute the values of |q1|, |q2|, x1, and x2 into the equation to find the positions x where the resultant force on q3 is zero.