Three charges are fixed to an xy coordinate system. A charge of +25C is on the y axis at y = +2.7 m. A charge of -16C is at the origin. Lastly, a charge of +66C is on the x axis at x = +2.7 m. Determine (a) the magnitude and (b) direction of the net electrostatic force on the charge at x = +2.7 m. Specify the direction as a positive angle relative to the +x axis.
To determine the net electrostatic force on the charge at x = +2.7 m, we need to calculate the individual forces exerted on it by the other charges and then find the vector sum of these forces.
Let's consider the charges one by one:
1. The charge at (+25C) on the y-axis at y = +2.7 m:
- The distance between this charge and the charge at x = +2.7 m is the hypotenuse of a right triangle.
- Using the Pythagorean theorem, we can calculate the distance as follows:
distance = sqrt((2.7 m)^2 + (2.7 m)^2) = sqrt(2 * (2.7 m)^2) = 2.7 m * sqrt(2).
- The magnitude of the force exerted by this charge on the charge at x = +2.7 m is given by Coulomb's Law:
force_1 = (k * |q1| * |q2|) / distance^2,
where k is the electrostatic constant (approximately 9 × 10^9 N m^2/C^2), |q1| is the magnitude of the charge at +25C, |q2| is the magnitude of the charge at x = +2.7 m.
2. The charge at (-16C) at the origin:
- The distance between this charge and the charge at x = +2.7 m is simply the x-coordinate of the latter.
- The magnitude of the force exerted by this charge on the charge at x = +2.7 m is given by Coulomb's Law:
force_2 = (k * |q1| * |q2|) / distance^2,
where distance is the x-coordinate of the charge at x = +2.7 m.
3. The charge at (+66C) on the x-axis at x = +2.7 m:
- The distance between this charge and the charge at x = +2.7 m is the difference between their y-coordinates.
- The magnitude of the force exerted by this charge on the charge at x = +2.7 m is given by Coulomb's Law:
force_3 = (k * |q1| * |q2|) / distance^2,
where distance is the difference between their y-coordinates.
Now, we need to calculate the net force by adding these forces as vectors. Since they are in different directions, we need to consider their components.
The x-component of the net force is given by:
Fx = force_2 - force_3.
The y-component of the net force is given by:
Fy = force_1.
Finally, the magnitude of the net force (Fnet) can be calculated using the Pythagorean theorem:
Fnet = sqrt(Fx^2 + Fy^2).
The direction of the net force relative to the +x axis can be determined using the arctan function:
θ = arctan(Fy / Fx).
By plugging in the appropriate values for the charges and distances, you can calculate the magnitude and direction of the net electrostatic force on the charge at x = +2.7 m.