1. The problem statement, all variables and given/known data
A point charge is placed at each corner of a square with side length a. The charges all have the same magnitude q. Two of the charges are positive and two are negative, as shown in the following figure.
The two positive charges are on top and the two negative charges are on the bottom
What is the magnitude of the net electric field at the center of the square due to the four charges in terms of q and a?
2. Relevant equations
E = k ( q / r^2), k = 9.0*10^9
3. The attempt at a solution
The distance from the middle to one of the charges a/(2^1/2)
The x-components cancel out, leaving only the y-components.
The electric field due to one of the charge to my guess is-
E1sinω = 9.0*10^9 ( 2q / a^2 ) * (1/ (2^1/2))
I assumed that each charge exerts the same electric field so the answer would\ be
4 * 9.0*10^9 ( 2q / a^2 ) * (1/ (2^1/2))
I am not sure what I did wrong.
Thanks so much
To find the magnitude of the net electric field at the center of the square, you need to consider the electric field contributions from each charge.
Let's break down the problem into steps:
Step 1: Calculate the electric field contribution from each charge.
Since the charges are at the corners of the square, the distance between the center of the square and each charge is a/√2 (as you correctly stated).
Using Coulomb's law, the electric field contribution from each charge is given by:
E = k * (q / r^2),
where k is the electrostatic constant (9.0 * 10^9 Nm^2/C^2), q is the charge magnitude, and r is the distance between the charge and the center of the square.
Hence, the electric field contribution from each charge is:
E1 = k * (q / (a/√2)^2) = k * (q * 2 / a^2) = 18.0 * 10^9 * (q / a^2).
Step 2: Determine the direction of the electric field contributions.
Since there are two positive charges and two negative charges, the electric field contributions from the positive charges will be in opposite directions compared to the contributions from the negative charges.
The contribution from one positive charge will point away from it, and the contribution from one negative charge will point towards it.
Therefore, the two positive charges will contribute electric fields that add up, while the two negative charges will contribute electric fields that also add up in the opposite direction.
Step 3: Calculate the net electric field.
Now, to find the magnitude of the net electric field, you need to consider the sum of the electric field contributions from all four charges.
Since the electric field contributions from the positive charges add up and the contributions from the negative charges add up but in the opposite direction, you have:
Net Electric Field = Electric Field from the positive charges - Electric Field from the negative charges.
Hence,
Net Electric Field = 2 * E1 - 2 * E1 = 4 * E1
Substituting the value of E1 calculated earlier, you get:
Net Electric Field = 4 * (18.0 * 10^9 * (q / a^2))
Therefore, the magnitude of the net electric field at the center of the square due to the four charges is 72.0 * 10^9 * (q / a^2).
Make sure to double-check your calculations and units to ensure accuracy.