Three charges are fixed to an x, y coordinate system. A charge of +20 µC is on the y axis at y = 2.7 m. A charge of -14 µC is at the origin. Last, a charge of +46 µC is on the x axis at x = +2.7 m. Determine the magnitude and direction of the net electrostatic force on the charge at x = +2.7 m. Specify the direction relative to the −x axis.
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To determine the magnitude and direction of the net electrostatic force on the charge at x = +2.7 m, we need to calculate the individual electrostatic forces between the charges and then add them vectorially.
1. Start by calculating the electrostatic force between the charge at x = +2.7 m and the charge at the origin (0,0).
The formula to calculate the electrostatic force between two charges is given by Coulomb's Law:
F = k * (q1 * q2) / r^2
Where:
F is the electrostatic force
k is the electrostatic constant (9 * 10^9 Nm^2/C^2)
q1 and q2 are the magnitudes of the charges
r is the distance between the charges
In this case, q1 is +2.7 µC and q2 is -14 µC. The distance between the charges is the distance from x = +2.7 m to the origin, which is also 2.7 m.
Plugging in these values into Coulomb's Law:
F1 = (9 * 10^9 Nm^2/C^2) * ((2.7 * 10^-6 C) * (-14 * 10^-6 C)) / (2.7 m)^2
Calculate F1 to find the electrostatic force between the charge at x = +2.7 m and the charge at the origin.
2. Next, calculate the electrostatic force between the charge at x = +2.7 m and the charge on the y-axis at y = 2.7 m.
Similar to step 1, use Coulomb's Law to calculate the electrostatic force. In this case, the charges are +2.7 µC and +20 µC. The distance between them is the distance from x = +2.7 m to y = 2.7 m, which is also 2.7 m.
F2 = (9 * 10^9 Nm^2/C^2) * ((2.7 * 10^-6 C) * (20 * 10^-6 C)) / (2.7 m)^2
Calculate F2 to find the electrostatic force between the charge at x = +2.7 m and the charge on the y-axis at y = 2.7 m.
3. Once we have calculated F1 and F2, we need to vectorially add them to find the net electrostatic force.
Since F1 and F2 act in different directions, we need to use vector addition. Add F1 and F2 by treating them as vectors with magnitude and direction.
To find the magnitude and direction of the net force, use the Pythagorean theorem and trigonometry. The magnitude of the net force is the square root of the sum of the squares of F1 and F2:
|F_net| = sqrt(F1^2 + F2^2)
The direction of the net force can be determined by finding the angle it makes with the negative x-axis. Use the inverse tangent (arctan) function:
θ = arctan(F2/F1)
Substitute the calculated values of F1 and F2 into the formulas above to find the magnitude and direction of the net electrostatic force on the charge at x = +2.7 m, relative to the -x axis.