Help with point charges and electric fields

[lyndan]

Homework Statement

So the question states: Four point charges, each of magnitude q, are located at the corners of a square with sides of length a. Two of the charges are +q and two are -q. The charges are arranged in one of the following two ways: (1) The charges alternate in sign (+q,-q,+q,-q) as you go around the square; (2) the top two corners of the square have positive charges (+q,+q) and the bottom two corners of the square have negative charges (-q,-q). (a) In which case will the electric field at the center of the square have the greatest magnitude? explain. (b) calculate the electric field at the center of the square for each of these two cases

The Attempt at a Solution

I already figured out that the electric field will have a greater magnitude in the second case, because the point charges in the first square will essentially cancel each other out? I can't figure out how to calculate the electric field at the center...I started using the E=klql/r^2 but I am just having trouble getting the answer...the first answer is zero and the answer to the second on is -4 2^1/2 kq / a^2 but so far I've been unable to get that

any help would be great thank you so much!

 

[tiny-tim]

...the first answer is zero and the answer to the second on is -4 2^1/2 kq / a^2 but so far I've been unable to get that

any help would be great thank you so much!

Hi lyndan!

Show us how far you've got, and then we'll be able to see where the difficulty is.

 

[baba89]

Hello,

Thank you for reaching out for help with your question on point charges and electric fields. It seems like you have a good understanding of the concept and have correctly identified that the electric field will have a greater magnitude in the second case.

To calculate the electric field at the center of the square, we can use the superposition principle. This states that the total electric field at a point is the sum of the electric fields due to each individual charge at that point.

In case 1, since the charges alternate in sign, the electric field at the center will be zero because the electric fields from the positive and negative charges will cancel each other out.

In case 2, we can use the formula for the electric field due to a point charge, E=kq/r^2, where k is the Coulomb's constant, q is the magnitude of the charge, and r is the distance from the charge to the point of interest. Since we have four point charges, we will have to calculate the electric field due to each one and then add them together.

Let's label the charges as q1, q2, q3, and q4. The electric field due to q1 and q3 will have the same magnitude and direction, and the electric field due to q2 and q4 will also have the same magnitude and direction. So, we can simplify the calculation by considering only the electric fields due to q1 and q2, and then multiplying the result by 2.

To find the distance from the charges to the center of the square, we can use the Pythagorean theorem. The distance from q1 to the center is a/2, and the distance from q2 to the center is a√2/2. Therefore, the electric field due to q1 and q2 can be calculated as:

E1 = kq/(a/2)^2 = 4kq/a^2
E2 = kq/(a√2/2)^2 = 2kq/a^2

Since E1 and E2 have the same direction, we can add them together to find the total electric field due to q1 and q2. Then, we can multiply the result by 2 to account for the electric fields due to q3 and q4.

Etotal = 2(E1 + E2) = 2(4kq/a^2 + 2kq/a^2