Three charges, +2.5 µC, -4.3 µC, and -6.0 µC, are located at (-0.20 m, 0.15 m), (0.50 m, -0.35 m), and (-0.42 m, -0.32 m), respectively. What is the electric field at the origin?
To find the electric field at the origin, we need to calculate the individual electric fields created by each charge and then combine them vectorially.
The electric field created by a point charge can be calculated using the formula:
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 between the charge and the point where the electric field is being calculated.
Let's calculate the electric field created by each charge:
1. For the +2.5 µC charge:
Q1 = +2.5 µC,
r1 = distance from (0.20 m, 0.15 m) to the origin = sqrt((0.20)^2 + (0.15)^2) = 0.25 m.
Using the formula:
E1 = k * (Q1 / r1^2) = (8.99 x 10^9 Nm^2/C^2) * (2.5 x 10^-6 C / (0.25 m)^2).
2. For the -4.3 µC charge:
Q2 = -4.3 µC,
r2 = distance from (0.50 m, -0.35 m) to the origin = sqrt((0.50)^2 + (-0.35)^2) = 0.61 m.
Using the formula:
E2 = k * (Q2 / r2^2) = (8.99 x 10^9 Nm^2/C^2) * (-4.3 x 10^-6 C / (0.61 m)^2).
3. For the -6.0 µC charge:
Q3 = -6.0 µC,
r3 = distance from (-0.42 m, -0.32 m) to the origin = sqrt((-0.42)^2 + (-0.32)^2) = 0.53 m.
Using the formula:
E3 = k * (Q3 / r3^2) = (8.99 x 10^9 Nm^2/C^2) * (-6.0 x 10^-6 C / (0.53 m)^2).
Now, let's calculate the electric field at the origin by vectorially adding these individual electric fields:
E = E1 + E2 + E3.
Calculate each component of the electric field separately. The x-component is the sum of the x-components of E1, E2, and E3, and the y-component is the sum of the y-components of E1, E2, and E3.
x-component:
Ex = E1x + E2x + E3x = (E1 * cos θ1) + (E2 * cos θ2) + (E3 * cos θ3),
where θ1, θ2, and θ3 are the angles that the vectors E1, E2, and E3 make with the x-axis.
y-component:
Ey = E1y + E2y + E3y = (E1 * sin θ1) + (E2 * sin θ2) + (E3 * sin θ3).
Finally, calculate the magnitude and direction of the resulting electric field using the following formulas:
|E| = sqrt(Ex^2 + Ey^2),
θ = arctan(Ey / Ex).
Substitute the values for E1, E2, and E3 obtained earlier to evaluate the electric field at the origin.
To find the electric field at the origin due to the three charges, we need to use the principle of superposition. According to this principle, the total electric field at a point in space is the vector sum of the individual electric fields produced by each charge.
The electric field due to a point charge can be calculated using Coulomb's law:
E = k * (q / r^2)
where E is the electric field, k is the Coulomb's constant (8.99 × 10^9 Nm^2/C^2), q is the charge, and r is the distance from the charge to the point where we want to calculate the electric field.
Let's calculate the electric field due to each charge individually:
For the +2.5 µC charge located at (-0.20 m, 0.15 m):
Distance from the charge to the origin (r1) = sqrt((-0.20 m)^2 + (0.15 m)^2)
Now, using Coulomb's law:
Electric field due to the +2.5 µC charge (E1) = (8.99 × 10^9 Nm^2/C^2) * (2.5 × 10^-6 C) / (r1)^2
Similarly, for the -4.3 µC charge located at (0.50 m, -0.35 m):
Distance from the charge to the origin (r2) = sqrt((0.50 m)^2 + (-0.35 m)^2)
Electric field due to the -4.3 µC charge (E2) = (8.99 × 10^9 Nm^2/C^2) * (-4.3 × 10^-6 C) / (r2)^2
And for the -6.0 µC charge located at (-0.42 m, -0.32 m):
Distance from the charge to the origin (r3) = sqrt((-0.42 m)^2 + (-0.32 m)^2)
Electric field due to the -6.0 µC charge (E3) = (8.99 × 10^9 Nm^2/C^2) * (-6.0 × 10^-6 C) / (r3)^2
Finally, we can find the net electric field at the origin by adding all the individual electric fields together:
Net electric field at the origin (E_net) = E1 + E2 + E3
Remember, the electric field is a vector, so don't forget to include the directions when adding them together.