Consider charges placed along an x-axis. If a charge of +17 nC is placed at x = 0.00 m and a charge of +32 nC is placed at x = 1.00 m, where would a third charge of -14.0 nC be in electrostatic equilibrium? Give the x-coordinate for your answer in meters and with three significant figures.
Obviously, the negative charge has to go between the positive charges.
let x be distance frm origin.
17q/x^2=14q/(1-x)^2
solve that quadratic for x.
To find the position of the third charge in electrostatic equilibrium, we need to make sure that the net electrostatic force acting on it is zero. The net electrostatic force is the vector sum of the forces exerted by the other charges on the third charge.
We can use Coulomb's law to calculate the electrostatic force:
F = k * |q1 * q2| / r^2
Where,
F is the magnitude of the electrostatic force,
k is the electrostatic constant (k = 9.0 x 10^9 N m^2/C^2),
q1 and q2 are the magnitudes of the charges,
r is the distance between the charges.
Let's calculate the forces exerted on the third charge by the other two charges.
The force exerted on the third charge by the charge at x = 0.00 m is:
F1 = k * |(-14.0 nC)(17 nC)| / (x2 - x1)^2
The force exerted on the third charge by the charge at x = 1.00 m is:
F2 = k * |(-14.0 nC)(32 nC)| / (x3 - x2)^2
Since the net electrostatic force on the third charge should be zero, F1 and F2 should have opposite directions and equal magnitudes. This means:
F1 = F2
Substituting the given values into the equations, we have:
k * |(-14.0 nC)(17 nC)| / (x2 - 0.00 m)^2 = k * |(-14.0 nC)(32 nC)| / (x3 - 1.00 m)^2
Simplifying the equation:
|(-14.0 nC)(17 nC)| / (x2 - 0.00 m)^2 = |(-14.0 nC)(32 nC)| / (x3 - 1.00 m)^2
Now, we can solve this equation to find the x-coordinate of the third charge.
|(-14.0 nC)(17 nC)| / (x2 - 0.00 m)^2 = |(-14.0 nC)(32 nC)| / (x3 - 1.00 m)^2
Rearranging the equation:
(x3 - 1.00 m)^2 = (x2 - 0.00 m)^2 * (|(-14.0 nC)(17 nC)| / |(-14.0 nC)(32 nC)|)
Taking the square root of both sides:
x3 - 1.00 m = ± (x2 - 0.00 m) * √(|(-14.0 nC)(17 nC)| / |(-14.0 nC)(32 nC)|)
Simplifying:
x3 = 1.00 m ± (x2 - 0.00 m) * √(|(-14.0 nC)(17 nC)| / |(-14.0 nC)(32 nC)|)
Substituting the given values:
x3 = 1.00 m ± (1.00 m - 0.00 m) * √(|(-14.0 nC)(17 nC)| / |(-14.0 nC)(32 nC)|)
Calculating the expression inside the square root:
√(|(-14.0 nC)(17 nC)| / |(-14.0 nC)(32 nC)|) ≈ 0.416
Substituting this value back into the equation:
x3 = 1.00 m ± (1.00 m - 0.00 m) * 0.416
Calculating:
x3 = 1.00 m ± 0.416 m
Therefore, the x-coordinate of the third charge in electrostatic equilibrium is approximately:
x3 ≈ 1.416 m or x3 ≈ 0.584 m
Rounding to three significant figures, the x-coordinate of the third charge in electrostatic equilibrium is 1.42 m or 0.584 m.