hard electric field problem!?

Question

There are charges arranged in a regular hexagon. The top three are positively charged, and going from left to right, have charges of 1uC, 2uC, and 3uC. The 3 bottom ones of the hexagon have charges going from left to right of 4uC, 5uC, and 6uC. Calculate the net electric field on a negative test charge placed at the left bottom-most corner (where the 6uC charge is at).

Answer 1

Ok, I am gonna give you the steps :

First, draw the regular hexagon : Now we have the six chargues, if we need to calculate the net electric field on a negative charge, then the electric field will be atractive, first we are gonna calculate, each electric field by every charge, so we need to find the distances betwees :

1 uC and 6 uC

2 uC and 6 uC

3 uC and 6 uC

4 uC and 6 uC

5 uC and 6 uC

Every side of the hexagon is " L "

then : distance from 1 uC and 6 uC = L
distance from 2 uC and 6 uC = L*sqrt(3)
distance from 3 uC and 6 uC = 2 L
distance from 4 uC and 6 uC = L*sqrt(3)
distance from 5 uC and 6 uC = L

So, we can find the electric field, the magnitude, then to find the net electric field, we will have to work with the components ok.

K = 9*10^9

Electric field by 1 uC = E1 = 9*10^9*10*-6 / L^2
E1 = 9*10^3 / L^2

Electric field by 2 uC = E2 = 9*10^9*2*10^-6 / 3*L^2
E2 = 6*10^3 / L^2

Electric field by 3 uC = E3 = 9*10^9*3*10^-6 / 4L^2
E3 = 27/4*10^3 / L^2

Electric field by 4 uC = E4 = 9*10^9*4*10^-6 / 3*L^2
E4 = 12*10^3 / L^2

Electric field by 5 uC = E5 = 9*10^9*5*10^-6 / L^2
E5 = 45*10^3 / L^2

Now : let's go back to the graphic :

E1, makes 60º with the horizontal ( to the negative side)

E2 , makes 90º with the horizontal

E3, makes 60º with the horizontal

E4, makes 30º with the horizontal

E5, is on the horizontal : 45*10^3 / L^2 i

Decomposing : E1, E2, E3, E4 :

E1 = 9*10^3 / L^2*cos(60) -i + 9*10^3 / L^2*sin(60) j

E1 = - 9*10^3 / 2L^2 i + 9*sqrt(3)10^3 / 2L^2 j

E2 = 6*10^3 / L^2 j

E3 = 27/4*10^3 / L^2*cos(60) i + 27/4*10^3 / L^2*sin(60) j

E3 = 27*10^3 / 8L^2 i + 27*sqrt(3)*10^3 / 8L^2 j

E4 = 12*10^3 / L^2*cos(30) i + 12*10^3 / L^2sin(30) j

E4 = 6*sqrt(3)*10^3 / L^2 i + 6*10^3 / L^2 j

E5 = 45*10^3 / L^2 i

E1 + E2 + E3 + E4 + E5 :

Net E = - 9*10^3 / 2L^2 i + 45*10^3 / L^2 i 6*sqrt(3)*10^3 / L^2 i + 27*10^3 / 8L^2 i + ( 6*10^3 / L^2 j + 27*sqrt(3)*10^3 / 8L^2 j + 6*10^3 / L^2 j + 9*sqrt(3)10^3 / 2L^2 j

E = 434*10^3 / 8L^2 i + 232*10^3 / 8L^2

Magnitude = 61.5*10^3 / L^2

That's is.

Hope that helped