A point charge q1=-8 is located at the orgin of a rectangular coordinate system. A second charge q2=6nC is located at a distance of 0.2m from the orgin on the positive x-axis, and a third charge q3=12nC is located at a distance of 0.2m on the positive y-axis. Find the magnitude and direction of the net electric force acting on the charge at the origin
F12=k•q1•q2/(r12)^2=1.08•10^-5 N (to the right)
F13= k•q1•q3/(r23)^2=2.16•10^-5 N (upwards)
F=sqroot(F12^2+F13^2)=2.4•10^-5 N
Direction of F respectively x-axis:
tan α=F13/F12= 2,
α = 63.4 degr
To find the magnitude and direction of the net electric force acting on the charge at the origin, we need to calculate the individual electric forces exerted by each of the other charges and then find their vector sum.
The formula to calculate the electric force between two charges is given by Coulomb's Law:
F = k * |q1 * q2| / r^2
Where:
F is the electric force,
k is the electrostatic constant (approximately 9 x 10^9 Nm^2/C^2),
|q1 * q2| is the magnitude of the product of the charges,
and r^2 is the square of the distance between the charges.
Step 1: Calculate the electric force between q1 and q2:
Here, q1 = -8C and q2 = 6nC = 6 * 10^-9 C
The distance between them is 0.2m.
Plugging the values into the formula:
F1 = k * |q1 * q2| / r^2
F1 = (9 * 10^9 Nm^2/C^2) * |-8 * 6 * 10^-9 C| / (0.2m)^2
Simplifying:
F1 = (9 * 10^9 Nm^2/C^2) * 48 * 10^-9 C / 0.04m^2
F1 = 4.32 * 10^-6 N
Step 2: Calculate the electric force between q1 and q3:
Here, q1 = -8C and q3 = 12nC = 12 * 10^-9 C
The distance between them is 0.2m.
Plugging the values into the formula:
F2 = k * |q1 * q3| / r^2
F2 = (9 * 10^9 Nm^2/C^2) * |-8 * 12 * 10^-9 C| / (0.2m)^2
Simplifying:
F2 = (9 * 10^9 Nm^2/C^2) * 96 * 10^-9 C / 0.04m^2
F2 = 10.8 * 10^-6 N
Step 3: Calculate the net electric force as a vector sum:
Since both F1 and F2 are on different axes, we can treat them as vector components.
The x-component of the net electric force is the sum of F1 and F2 in the x-direction: Fx = F1 + F2 = 4.32 * 10^-6 N + 10.8 * 10^-6 N = 15.12 * 10^-6 N
The y-component of the net electric force is the sum of F1 and F2 in the y-direction: Fy = F2 = 10.8 * 10^-6 N
The magnitude of the net electric force is given by the Pythagorean theorem: |F| = sqrt(Fx^2 + Fy^2) = sqrt((15.12 * 10^-6 N)^2 + (10.8 * 10^-6 N)^2)
Calculating:
|F| = sqrt((228.1 * 10^-12 N^2) + (116.64 * 10^-12 N^2))
|F| = 13.67 * 10^-6 N
The direction of the net electric force can be found using trigonometry.
Theta = arctan(Fy / Fx) = arctan((10.8 * 10^-6 N) / (15.12 * 10^-6 N))
Calculating:
Theta = arctan(0.7143) ≈ 35.78 degrees
Therefore, the magnitude of the net electric force acting on the charge at the origin is approximately 13.67 * 10^-6 N, and its direction is approximately 35.78 degrees with respect to the positive x-axis.