1. Three point charges are placed at the following (x, y) coordinates: charge +4.0 µC at (0.0, 0.5 m), charge +1.0
µC at (0.2 m, 0.0) and charge -5.0 µC at (0.2m, 0.5m). Calculate the net electric field at the origin due to these three point charges.
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To calculate the net electric field at the origin due to these three point charges, we need to calculate the individual electric fields created by each charge and then add them up vectorially.
The electric field created by a point charge can be calculated using Coulomb's law:
E = k * Q / r^2
where E is the electric field, k is the electrostatic constant (8.99 x 10^9 Nm^2/C^2), Q is the charge, and r is the distance from the charge.
Let's start by calculating the electric field created by the +4.0 µC charge at (0.0, 0.5m). Since the origin is at (0, 0), the distance from this charge to the origin is 0.5m. Plugging the values into the equation:
E1 = (8.99 x 10^9 Nm^2/C^2) * (4.0 x 10^-6 C) / (0.5m)^2
Calculating this, we get E1 = 71920000 N/C, in the downward direction (since the charge is positive).
Next, let's calculate the electric field created by the +1.0 µC charge at (0.2m, 0.0). The distance from this charge to the origin is 0.2m. Plugging the values into the equation:
E2 = (8.99 x 10^9 Nm^2/C^2) * (1.0 x 10^-6 C) / (0.2m)^2
Calculating this, we get E2 = 224750000 N/C, in the upward direction (since the charge is positive).
Finally, let's calculate the electric field created by the -5.0 µC charge at (0.2m, 0.5m). The distance from this charge to the origin is the diagonal distance, which can be calculated using the Pythagorean theorem:
distance = √((0.2m)^2 + (0.5m)^2) = √(0.04m^2 + 0.25m^2) = √(0.29m^2) = 0.5385m
Plugging the values into the equation:
E3 = (8.99 x 10^9 Nm^2/C^2) * (-5.0 x 10^-6 C) / (0.5385m)^2
Calculating this, we get E3 = -58075308 N/C, in the upward direction (since the charge is negative).
To find the net electric field at the origin, we need to add these three electric fields vectorially. Since E1 and E3 are in the opposite directions, they will partially cancel each other out. Adding the magnitudes of E1, E2, and E3, we get:
|E1| + |E2| + |E3| = 71920000 N/C + 224750000 N/C + 58075308 N/C = 353925308 N/C
Now, to add the directions, we need to subtract the magnitude of the smaller electric field from the magnitude of the larger electric field, and keep the direction of the larger electric field. In this case, |E2| is larger than |E1|, but E2 and E3 have opposite directions, so we subtract |E1| from |E2|:
|E2| - |E1| = 224750000 N/C - 71920000 N/C = 152830000 N/C
Therefore, the net electric field at the origin due to these three point charges is approximately 152830000 N/C in the upward direction.