Four equal charged +Q are placed at the corners of a square of side a. Find the electric field strength at the midpoint of one side. Express your answer in terms of Q, a, and the constant k
To find the electric field strength at the midpoint of one side of the square, we can use the principle of superposition.
First, let's consider one of the charges at the corner of the square. The distance between this charge and the midpoint of one side is half the length of a side, which is a/2.
The electric field strength of this individual charge at the midpoint can be calculated using the formula:
E = k * (q / r^2)
where E is the electric field strength, k is the electrostatic constant (k = 9 x 10^9 N m^2/C^2), q is the charge, and r is the distance between the charges.
In this case, we have four charges at the corners of the square, all with the same magnitude, Q. So, the net electric field at the midpoint will be the vector sum of the electric fields due to these individual charges.
Since the charges are located at the corners of a square, two of these charges will contribute a component of the electric field that is vertical (parallel to the side of the square) at the midpoint, and the other two charges will contribute a horizontal component.
Considering the vertical component first, due to symmetry, we can observe that the electric field vectors from the charges, located diagonally opposite to each other, will cancel each other out.
So, the net vertical component of the electric field at the midpoint is zero.
Now, let's calculate the net horizontal component of the electric field at the midpoint due to the two remaining charges. Since these charges are equidistant, we can find this contribution by considering only one of the charges.
The horizontal distance between one of the charges and the midpoint is a/2, and the horizontal component of the electric field due to one charge is given by:
E' = k * (Q / (a/2)^2)
Multiplying this value by 2 (since we have two charges contributing to the horizontal component), we get:
E_horizontal = 2 * k * (Q / (a/2)^2)
Simplifying further, we have:
E_horizontal = 2 * k * (Q / (a^2 / 4))
E_horizontal = 8 * k * (Q / a^2)
Therefore, the electric field strength at the midpoint of one side of the square is 8kQ/a^2, where k is the electrostatic constant, Q is the charge, and a is the side length of the square.