To find the electric field at the origin, we need to calculate the electric field contribution from each charge and then sum them up vectorially.
The electric field due to a point charge at a given location can be calculated using Coulomb's Law:
E = k * (Q / r^2) * u
where E is the electric field, k is the electrostatic constant (9.0 x 10^9 N m^2/C^2), Q is the charge, r is the distance between the charge and the point of interest, and u is the unit vector pointing from the charge to the point of interest.
Let's calculate the electric field due to each charge:
Charge 1: +2.5 μC at (0.20 m, 0.15 m)
- The distance from charge 1 to the origin is:
r1 = sqrt((0.20 m)^2 + (0.15 m)^2) = 0.25 m
- The unit vector from charge 1 to the origin is:
u1 = (0.20 m / r1, 0.15 m / r1) = (0.80, 0.60)
- The electric field due to charge 1 is:
E1 = k * (2.5 x 10^-6 C / (0.25 m)^2) * (0.80, 0.60) = (k * 8.0 x 10^-2) * (0.80, 0.60)
Charge 2: -4.8 μC at (0.50 m, -0.35 m)
- The distance from charge 2 to the origin is:
r2 = sqrt((0.50 m)^2 + (-0.35 m)^2) = 0.61 m
- The unit vector from charge 2 to the origin is:
u2 = (0.50 m / r2, -0.35 m / r2) = (0.82, -0.57)
- The electric field due to charge 2 is:
E2 = k * (-4.8 x 10^-6 C / (0.61 m)^2) * (0.82, -0.57) = (k * -13.06) * (0.82, -0.57)
Charge 3: 6.3 μC at (-0.42 m, -0.32 m)
- The distance from charge 3 to the origin is:
r3 = sqrt((-0.42 m)^2 + (-0.32 m)^2) = 0.53 m
- The unit vector from charge 3 to the origin is:
u3 = (-0.42 m / r3, -0.32 m / r3) = (-0.79, -0.61)
- The electric field due to charge 3 is:
E3 = k * (6.3 x 10^-6 C / (0.53 m)^2) * (-0.79, -0.61) = (k * 22.8) * (-0.79, -0.61)
Now, let's sum up the contributions from each charge vectorially:
E_total = E1 + E2 + E3
By plugging in the values for k and calculating the vector sum of E_total, we can find the electric field at the origin.