Help with a physics question? How did they get this number?

Question

*Point charges of -8.20 micro-coulombs are placed at each corner of a square that is 0.250m on a side. Find the magnitude of the electric field at the midpoint of any of the sides.

Im talking about the bottom right equation. How did they get 8/5? also, what is the value of a^2?

Basically, the bottom right equation is the left step, just need to plug in numbers.

Argent · Answer

Q is the magnitude of each charge, -8.20 µC.

"a" is the length of the side of the square arrangement of charges. Its exact value is not computable here, but that does not matter.
a/2 is 1/2 the length of the side, so d is the hypotenuse of a right triangle with legs equal to a and a/2. By the Pythagorean relation, d² = a² + (a/2)² = a² + a²/4 = (5/4)a².

d², expressed as (5/4)a², is the denominator of the last equation:
E = 2(kQ/((5/4)a²))·cos(26.6°). Since there is a fraction (5/4) in the denominator, it can be inverted and multiplied by the numerator, as in A/(B/C) = A·(C/B).

Inverting the fraction in this way, we can rewrite the equation as
2(4kQ/(5a²))·cos(26.6°).
Moving the 4 outside the parentheses (multiplying it by the 2), we get
8(kQ/(5a²))·cos(26.6°).
Moving the 5 outside the parentheses (keeping it as the denominator, under the 8), we get
(8/5)(kQ/a²)·cos(26.6°).

? · Answer

thanks so much!

Help with a physics question? How did they get this number?

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