A charge of -6.5 micro-Coulombs is placed at the origin of a coordinate system. Another charge of -7.6 micro-Coulombs is placed at x = +0.29 m, y = +0.17 m. A third charge of +14 micro-Coulombs is placed at x = -0.29 m, y = 0 m. What is the magnitude of the total electric field on the charge at the the point x = 0 m, y = +0.17 m in units of 1 x 106 N/C?
To find the magnitude of the total electric field at a given point, we need to calculate the electric field produced by each individual charge at that point and then sum them together. The formula for the electric field due to a point charge is given by:
E = k * |Q| / r^2
Where:
- E is the electric field
- k is the Coulomb's constant (9 x 10^9 Nm^2/C^2)
- Q is the charge
- r is the distance between the charge and the point
Let's calculate the electric field due to each charge and then sum them up:
1. Electric field due to the charge at the origin:
E1 = k * |Q1| / r1
= (9 x 10^9 Nm^2/C^2) * |(-6.5 x 10^-6 C)| / (0.17 m)
= (9 x 10^9 Nm^2/C^2) * (6.5 x 10^-6 C) / (0.17 m)
= 3.5 x 10^5 N/C
2. Electric field due to the charge at (0.29 m, 0.17 m):
E2 = k * |Q2| / r2
= (9 x 10^9 Nm^2/C^2) * |(-7.6 x 10^-6 C)| / [(0.29 m)^2 + (0.17 m)^2]^0.5
= (9 x 10^9 Nm^2/C^2) * (7.6 x 10^-6 C) / [(0.29 m)^2 + (0.17 m)^2]^0.5
= 1.4 x 10^5 N/C
3. Electric field due to the charge at (-0.29 m, 0 m):
E3 = k * |Q3| / r3
= (9 x 10^9 Nm^2/C^2) * |(14 x 10^-6 C)| / [(0.29 m + 0.29 m)^2 + (0 m)^2]^0.5
= (9 x 10^9 Nm^2/C^2) * (14 x 10^-6 C) / [(0.58 m)^2]^0.5
= 1.3 x 10^5 N/C
Now, let's sum up the electric fields:
E_total = E1 + E2 + E3
= 3.5 x 10^5 N/C + 1.4 x 10^5 N/C + 1.3 x 10^5 N/C
= 6.2 x 10^5 N/C
Therefore, the magnitude of the total electric field on the charge at the point (0 m, 0.17 m) is 6.2 x 10^5 N/C.