A point charge of -4 mu or micro CC is located at x = 4 m, y = -2 m. A second point charge of 12 mu or micro CC is located at x = 1 m, y = 4 m.

(a) Find the magnitude and direction of the electric field at x = -1 m, y = 0.


I got 4545.69 N/C by using this equation E= kQ/r^2 (vector r) and finding EQ1 and EQ2 then finding their x and y components and adding those then taking the magnitude. but my answer is wrong.

You are on the right track; however, you may have made a mistake with calculating the distances or summing the components. Let's do the calculation again.

(a) To find the electric field at point P(-1, 0), we need to find the fields due to each charge separately and then sum the contributions.

Let's denote the point where the -4 μC charge is located as A(4, -2) and the point where the 12 μC charge is located as B(1, 4).

The distance vector from A to P is r1 = <-5, 2> and from B to P r2 = <-2, -4>.

The magnitudes of the distance vectors are |r1| = sqrt((-5)^2 + 2^2) = sqrt(29) and |r2| = sqrt((-2)^2 + (-4)^2) = sqrt(20).

Now, let's find the electric fields due to each charge at point P:

E1 = k * q1 / |r1|^2 * r1 = (9 × 10^9 N m^2/C^2) * (-4 × 10^-6C) / (29 m^2) * <-5, 2>
E1 = <-60, 24> N/C

E2 = k * q2 / |r2|^2 * r2 = (9 × 10^9 N m^2/C^2) * (12 × 10^-6C) / (20 m^2) * <-2, -4>
E2 = <108, 216> N/C

Now, let's find the total electric field at point P by summing the components of E1 and E2:

E_total = E1 + E2 = (<-60 + 108, 24 + 216>) = (48, 240) N/C

The magnitude and direction of the total electric field are:

|E_total| = sqrt(48^2 + 240^2) ≈ 246.6 N/C

tan(theta) = 240/48 => theta ≈ 78.7° (above the negative x-axis in the second quadrant)

Therefore, the magnitude and direction of the electric field at point P are approximately 246.6 N/C and 78.7° above the negative x-axis.

To find the magnitude and direction of the electric field at the point (x = -1 m, y = 0), you need to calculate the electric field contributions from both point charges.

Let's calculate the electric field contribution from the first point charge (-4 μC) located at (4 m, -2 m).

1. Calculate the distance (r1) between the first point charge and the point where we want to find the electric field:
r1 = √((x1 - x)^2 + (y1 - y)^2)
= √((4 m - (-1 m))^2 + (-2 m - 0)^2)
= √((5 m)^2 + (-2 m)^2)
= √(25 m^2 + 4 m^2)
= √(29 m^2)
= 5.39 m

2. Calculate the electric field contribution (E1) from the first point charge using the formula:
E1 = k * Q1 / r1^2
where k is the electrostatic constant (k = 8.99 x 10^9 N·m^2/C^2) and Q1 is the charge of the first point charge (-4 μC = -4 x 10^-6 C).
E1 = (8.99 x 10^9 N·m^2/C^2) * (-4 x 10^-6 C) / (5.39 m)^2
E1 = -1.60 x 10^6 N/C (approximately)

Now, let's calculate the electric field contribution from the second point charge (12 μC) located at (1 m, 4 m).

3. Calculate the distance (r2) between the second point charge and the point where we want to find the electric field:
r2 = √((x2 - x)^2 + (y2 - y)^2)
= √((1 m - (-1 m))^2 + (4 m - 0)^2)
= √((2 m)^2 + (4 m)^2)
= √(4 m^2 + 16 m^2)
= √(20 m^2)
= 4.47 m

4. Calculate the electric field contribution (E2) from the second point charge using the formula:
E2 = k * Q2 / r2^2
where Q2 is the charge of the second point charge (12 μC = 12 x 10^-6 C).
E2 = (8.99 x 10^9 N·m^2/C^2) * (12 x 10^-6 C) / (4.47 m)^2
E2 = 3.03 x 10^5 N/C (approximately)

5. Add the two electric field contributions to get the total electric field at the point (x = -1 m, y = 0):
E = E1 + E2
E = -1.60 x 10^6 N/C + 3.03 x 10^5 N/C
E = -1.29 x 10^6 N/C (approximately)

The magnitude of the electric field at (x = -1 m, y = 0) is approximately 1.29 x 10^6 N/C, and the direction is negative (pointing towards the negative x-axis).

To find the magnitude and direction of the electric field at a given point, you need to consider the contributions from both point charges.

Let's start by defining the variables:

Q1 = -4 μC (charge of the first point charge)
Q2 = 12 μC (charge of the second point charge)
r1 = distance from the first point charge to the point of interest
r2 = distance from the second point charge to the point of interest
k = Coulomb's constant (approximately 9 × 10^9 N m^2/C^2)

In this case, the point of interest is located at coordinates (x = -1 m, y = 0).

Step 1: Calculate the electric field due to Q1 at the point of interest:

First, calculate the vector r1 as follows:
r1 = (x1 - x, y1 - y) = (4 - (-1), -2 - 0) = (5, -2)

Then, calculate the magnitude of r1:
|r1| = √(5^2 + (-2)^2) ≈ √29 m

Now, calculate the electric field due to Q1:
E1 = k * Q1 / r1^2
= (9 × 10^9 N m^2/C^2) * (-4 × 10^-6 C) / (29 m)^2

Step 2: Calculate the electric field due to Q2 at the point of interest:

Perform the same calculations as for Q1, but using the coordinates and charge of Q2:
r2 = (x2 - x, y2 - y) = (1 - (-1), 4 - 0) = (2, 4)
|r2| = √(2^2 + 4^2) = √20 m

E2 = k * Q2 / r2^2
= (9 × 10^9 N m^2/C^2) * (12 × 10^-6 C) / (20 m)^2

Step 3: Calculate the net electric field at the point of interest:

To find the net electric field, add the electric fields due to Q1 and Q2 together vectorially:

E_net = E1 + E2

Step 4: Calculate the magnitude and direction of the net electric field:

To find the magnitude of the net electric field, calculate the magnitude of E_net:

|E_net| = √(E_net.x^2 + E_net.y^2)

To find the direction of the net electric field, calculate the angle it makes with the positive x-axis using trigonometry:

θ = arctan(E_net.y / E_net.x)

Now, substitute the values of E1 and E2 into E_net, calculate |E_net| and θ to obtain the final answer for the magnitude and direction of the electric field at the given point.