Consider three charges at points 𝑥1, 𝑦1 = (2.6 m, 1.2 m), 𝑥2, 𝑦2 = (3.7 m, 7.8 m), 𝑥3, 𝑦3 = (9.3 m, 5.2 m) with charges 𝑞1 = 7 nC, 𝑞2 = −9.6 nC, 𝑞3 = 4.7 nC. What is the magnitude of the electric field at the origin (𝑥 = 0, 𝑦 = 0) in N/C? (10 marks)

To solve this problem, we need to use the formula for the electric field due to a point charge, which is given by:

E = k * |q| / r^2

where:

• E is the electric field,
• k is Coulomb's constant (9 * 10^9 N m^2/C^2),
• q is the charge, and
• r is the distance from the charge to the point of interest.

We calculate the electric field due to each charge at the origin and then add them up vectorially.

1. For q1: The distance r1 from q1 to the origin is sqrt((2.6 m)^2 + (1.2 m)^2) = 2.8 m. So, the electric field E1 due to q1 at the origin is E1 = k * |q1| / r1^2 = 9 * 10^9 * 7 * 10^-9 / (2.8)^2 = 0.9 N/C.

2. For q2: The distance r2 from q2 to the origin is sqrt((3.7 m)^2 + (7.8 m)^2) = 8.6 m. So, the electric field E2 due to q2 at the origin is E2 = k * |q2| / r2^2 = 9 * 10^9 * 9.6 * 10^-9 / (8.6)^2 = 1.2 N/C.

3. For q3: The distance r3 from q3 to the origin is sqrt((9.3 m)^2 + (5.2 m)^2) = 10.4 m. So, the electric field E3 due to q3 at the origin is E3 = k * |q3| / r3^2 = 9 * 10^9 * 4.7 * 10^-9 / (10.4)^2 = 0.4 N/C.

Finally, we add up the electric fields due to each charge to get the total electric field at the origin. Since the charges are not all in the same direction, we need to add them up vectorially, which involves calculating the x and y components of each electric field and then adding them up separately.

The x component of the electric field due to a charge is E * cos(theta), and the y component is E * sin(theta), where theta is the angle between the direction of the electric field and the x-axis.

After calculating the x and y components of each electric field and adding them up, we can find the magnitude of the total electric field at the origin using the Pythagorean theorem:

E_total = sqrt((sum of x components)^2 + (sum of y components)^2)

This will give the magnitude of the electric field at the origin in N/C.