Electric Field at the Center of a Triangle

[Brandone]

Homework Statement

Given an equilateral triangle with sides 20-cm, with point charges at each point:
q₁ at the top with a charge of +4.0uM,
q₂ at the bottom right with a charge of -4.0uM, and
q₃ at the bottom left with a charge of +4.0uM.

k is assumed to be exactly 9.00x10⁹ N-m²/c²

Question: What is the electric field at the center of the triangle?

Homework Equations

E = (kq)/r²

The Attempt at a Solution

First and foremost, I employed a little geometry find the center of the triangle:

r = sqrt( (0.20-m)² - (0.10-m)² ) / 2
r = 0.866-m, rounded to
r = 0.90-m

E₁ = (9.00x10⁶ N-m²/c²)(4.0x10^-6C) / (0.09-m)²
E₁ = 4.4x10⁶ N/C

The magnitudes of all three charges are equivalent, so:

E₁ = E₂ = E₃

So the x and y components would be:

E_x = (4.4x10⁶ N/C)cos30° + (4.4x10⁶ N/C)cos330°
E_x = 7.6 x 10⁶ N/C

E_y = (4.4x10⁶ N/C)sin270° + (4.4x10⁶ N/C)sin30° + (4.4x10⁶ N/C)sin330°
E_y = -4.4x10⁶ N/C

E = sqrt[ (7.6 x 10⁶ N/C)² - (4.4x10⁶ N/C)² ]
E = 6.2 x 10⁶ N/C

So, my answer is 6.2 x 10⁶ N, but the book answer key shows 5.4 x 10⁶ N/C. That is quite a significant difference, but I can't seem to find where I'm making a mistake. It's driving me mad!

 

[ehild]

r = sqrt( (0.20-m)² - (0.10-m)² ) / 2
r = 0.866-m, rounded to
r = 0.90-m

Check your geometry. How far are the corners of an equilateral triangle from the centre?

You rounded r too much, just at the beginning of your calculations.
The value of r has got 4% error because of rounding. It is squared, that doubles the error to 8% which increases further during the rest of the calculations.

You must not round so much during the calculations. Keep 3-4 significant digits.

You have r² in the formula for the electric field intensity. So you do not need the square root at all and.


ehild

 

[Brandone]

I must be daft!

And I made a typo, it's 0.09 and not 0.90. But that's beyond the point!

The distance from one corner of an equilateral triangle is equal to a hypotenuse made from a right triangle with one leg half the length of the sides of the triangle and the other leg equal to 1/3 the value of the distance from one corner to the midpoint of the opposing side!

SO

r² = [ sqrt( (0.20m)² - (0.10m)² ) / 3 ]² + (0.1m)²
r² = 0.0133 m

Thank you very much, ehild! :-D