Four point charges have the same magnitude of 2.62 10-12 C and are fixed to the corners of a square that is 3.80 cm on a side. Three of the charges are positive and one is negative. Determine the magnitude of the net electric field that exists at the center of the square.
Symettry. Two opposite charges must be +, so they cancel E since opposite directions. tHe other two add, so E=k2q/distance^2
where distance is 0ne half of the diagonal.
To determine the magnitude of the net electric field at the center of the square, we can use the principle of superposition.
The electric field due to a point charge at a given point is given by Coulomb's law equation:
E = k * Q / r^2
Where:
- E is the electric field strength
- k is the electrostatic constant (approximately 9 x 10^9 Nm^2/C^2)
- Q is the charge magnitude
- r is the distance between the charge and the point where the electric field is measured
Since the problem states that three of the charges are positive and one is negative, we can assume that the charges are arranged in a way that the net electric field at the center will be zero. This is because the positive charges will contribute to the electric field in one direction, while the negative charge will contribute in the opposite direction, canceling out the effects.
To calculate the net electric field, we will take the electric field due to each individual charge at the center of the square and add them up. Since the charges are placed at the corners of the square and are equidistant from the center, the distances between each charge and the center are equal.
Let's assume the side length of the square is "a" and the distance from the center to one of the charges is "r." In this case, since all sides of the square have the same length, a = 3.80 cm and r = a / sqrt(2).
The magnitude of the electric field due to a point charge Q at the center of the square is:
E = k * Q / r^2
For the positive charges, the electric field direction is outward, away from the charge, and for the negative charge, the electric field direction is inward, towards the charge.
Therefore, the magnitude of the net electric field at the center of the square is:
E_net = E_positive1 + E_positive2 + E_positive3 - E_negative
Substituting the given values:
E_net = (k * Q / r^2) + (k * Q / r^2) + (k * Q / r^2) - (k * Q / r^2)
E_net = k * Q (1/r^2 + 1/r^2 + 1/r^2 - 1/r^2)
Since all charges have the same magnitude, we can simplify this to:
E_net = 4 * (k * Q / r^2)
Substituting the known values:
E_net = 4 * (9 x 10^9 Nm^2/C^2 * 2.62 x 10^(-12) C / (a / sqrt(2))^2)
E_net = 4 * (9 x 10^9 Nm^2/C^2 * 2.62 x 10^(-12) C / (0.038 m / sqrt(2))^2)
E_net = 4 * (9 x 10^9 Nm^2/C^2 * 2.62 x 10^(-12) C / (0.0269 m)^2)
E_net ≈ 1.44 x 10^8 N/C
Therefore, the magnitude of the net electric field at the center of the square is approximately 1.44 x 10^8 N/C.