Question

Find the magnitude of the net electric force exerted on a charge Q, located at the center of the square, for the following arrangement of charge: the two positive charges are on the top corners, and the two negative charges are on the bottom corners.

Express your answer in terms of the variables q, Q, a, and appropriate constants.

Answer 1

F = 5.09 10¹⁰ q Q / a²

Step-by-step explanation:

The electric force is given by Coulomb's law

F = k q₁ q₂ / r²

Let's apply this equation to our case, we have four force, the charge in the singer is positive Q, so the two upper charges have repulsive forces directed in the diagonals of the square and the two lower charges that are negative, have attractive forces directed at the length of the diagonals.

To find the total force we know how to use Newton's second law

On the x-axis we can see that all forces will cancel out

On the Y axis, all forces have the same direction, therefore the four forces are added.

We can find the value of one of them

The distance from the corner to the center of the square is

r² = (a / 2)² + (a / 2)² = 2 (a / 2)²

r² = a² / 2

Force is worth

F = k q Q / r²

F = k q Q 2 / a²

Let's use trigonometry to find the vertical component of this force

sin45 =
`F_(y)` / F

`F_(y)` = F sin 45

`F_(y)` = k q Q 2 / a² 07071

The total force is the

F = 4
`F_(y)`

F = k q Q / a2 8 0.7071

F = 5.09 10¹⁰ q Q / a²