Calculating total Coulomb force vector ?

[starstruck_]

Homework Statement

Consider a configuration consisting one +q charge ( upper right) and three −q charges, arranged in a square.

Side lengths = d.

Calculate the total F force vector acting on charge +q.

Homework Equations

Vector form of culomb’s force
F=( kq1q2/r^2) rhat

(rhat for unit vector - I’m on my phone so I can’t really tyupe it out properly, sorry)

The Attempt at a Solution

Split into 3 Force vector (between +q and each charge ). F1 is horizontal, F2 is the diagonal force vector, F3 is the vertical force vector.

F1 = (-ke^2/r^2)rhat

r= d
rhat= (ihat)
Or is it rhat= d( ihat )?

Anyways,
F1= ((-ke^2)/d^2))(ihat)

F2 = (-ke^2/r^2)(rhat)
r = sqrt(d^2+d^2)
r= sqrt(2)d
rhat = r vector/|r vector|
Assuming tail at origin
r vector = <d,d>
rhat= d(ihat)+d(jhat)/(sqrt(2)d)

F2= (-ke^2/(2d^2))(d(ihat)/sqrt(2)d)+ (-ke^2/(2d^2))(d(jhat)/sqrt(2)d)

F2 =(-ke^2/(2sqrt(2)d^3))(d(ihat))+(-ke^2/(2sqrt(2)d^3))(d(jhat))

F3 = (-ke^2/r^2)(rhat)

r= d
rhat= jhat or is it rhat = d(jhat)?

F3= (-ke^2/d^2)(jhat)

And then I would just add all of the components together.

I’m just wondering if I did this rhat business correctly ?

It said to use the vector form of the Coulomb force, so I tried- I’m not used to working with forces in this manner.

((Also sorry if it’s really hard to understand what I did, I can try to write it down and post a picture of my work if possible)).

 

[rude man]

Your basic approach is OK.

It's convenient in this forum to use bold letters to convey vectors.
So r_hat = r, i_hat = i, j_hat = j, r =r cosθ i + r sinθ j and F = F_x i + F_y j.

When you introduce r you are doing coordinate system switching between cartesian (x,y) and polar (r,θ) coordinates. But this is not necessary. You can just stick with cartesian. So for example F₂ = kq/d² i + kq/d² j and so on for F₁ and F₃. Then just add all the x and y components separately to get the net force in the i and j directions. Note that you don't calculate r² =2d²separately. At the end you can still compute r (and θ) if you want.