Three point charges lie along a circle of radius r at angles of 30°, 150°, and 270° as shown in the figure below. Find a symbolic expression for the resultant electric field at the center of the circle.

wouldn't all the vectors add to zero?

To find the resultant electric field at the center of the circle due to the three point charges, you can use the principle of superposition. This principle states that the total electric field at a point is the vector sum of the individual electric fields produced by each point charge.

In this case, let's assume the magnitude of each point charge is Q. The symbolic expression for the resultant electric field at the center of the circle, denoted as E(total), can be found using the following steps:

Step 1: Calculate the electric field produced by each point charge at the center of the circle.
The electric field at the center of the circle due to a point charge can be calculated using the formula:
E = (k * Q) / r^2
where k is the electrostatic constant (approximately 9 × 10^9 N m^2/C^2) and r is the distance from the charge to the center of the circle.

For the angle 30°, the distance from the charge to the center of the circle is r. So, the electric field at the center of the circle due to this charge is:
E1 = (k * Q) / r^2

For the angle 150°, the distance from the charge to the center of the circle is also r. So, the electric field at the center of the circle due to this charge is:
E2 = (k * Q) / r^2

For the angle 270°, the distance from the charge to the center of the circle is 2r (as it lies on the diameter). So, the electric field at the center of the circle due to this charge is:
E3 = (k * Q) / (2r)^2 = (k * Q) / 4r^2

Step 2: Calculate the resultant electric field.
To find the resultant electric field at the center of the circle, we need to add up the individual electric fields:

E(total) = E1 + E2 + E3

Substituting the values we derived earlier, the symbolic expression for the resultant electric field at the center of the circle is:
E(total) = (k * Q) / r^2 + (k * Q) / r^2 + (k * Q) / 4r^2

Combining like terms:
E(total) = (2k * Q) / r^2 + (k * Q) / 4r^2

Therefore, the symbolic expression for the resultant electric field at the center of the circle is:
E(total) = (8k * Q) / (4r^2) + (k * Q) / 4r^2

Simplifying further, we get:
E(total) = (9k * Q) / (4r^2)