The electric field at the center of a square

[Jay9313]

http://helios.augustana.edu/~dr/203/probs/Set%201.pdf
Scroll down to #11 for a picture.

So I kind of understand the math. I am a little confuse though. So why is a divided by sqrt(2)?
If this is a vector, then the center is really (a(sqrt(2)))/2 and the x component would be divided by sqrt(2) which makes the equation equal a/2. Can someone explain this? Is my math wrong?

 

[Spinnor]

The diagonal distance from corner to corner is a*2^.5 and the distance from corner to the center is half of that,

a*2^.5/2 = a/2^.5

Yes?

 

[Redbelly98]

The distance from the center to any corner is $a/\sqrt{2}$, that's why a is divided by sqrt(2).

Taking the x-component is accounted for by the cos(45) in the expression.

 

[Jay9313]

I'm not sure why it's a/sqrt(2). I do know that (a(sqrt(2))/2 is the exact same number. I don't know why.

Basically, I found the length of the diagonal, a(sqrt(2)) and divided it by 2. It's the same as a/sqrt(2)

 

[rwisz]

The electric field at the center of a square can be calculated by using the superposition principle, which states that the total electric field at a point is the sum of the individual electric fields due to each charge in the system. In this case, we have four charges at the corners of the square, each with the same magnitude and direction of electric field.

To calculate the electric field at the center, we can use Coulomb's law, which states that the magnitude of the electric field at a point is directly proportional to the magnitude of the charge and inversely proportional to the square of the distance between the point and the charge.

In the case of a square, the distance from the center to each corner is a/√2, as shown in the diagram. This is because the diagonal of a square is √2 times the length of one side. So, when we plug this into Coulomb's law, we get a/√2 in the denominator, which explains why the equation is divided by √2.

As for the x component, you are correct that the center of the square is actually (a√2)/2. However, when we are calculating the x component of the electric field, we only consider the distance in the x direction, which is a/2. This is why the equation simplifies to a/2 for the x component.

In summary, the reason for dividing by √2 in the equation for the electric field at the center of a square is due to the geometry of the square and the application of Coulomb's law. Your math is correct, and I hope this explanation helps to clarify any confusion.