Calculating total Coulomb force vector ?

In summary, the homework statement is to calculate the total force acting on a charge located at the upper right corner of a square.
  • #1
starstruck_
185
8

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)).
 
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  • #2
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 = Fx i + Fy 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.
 
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1. What is the formula for calculating total Coulomb force vector?

The formula for calculating total Coulomb force vector is F = (k * Q1 * Q2) / r^2, where F is the total force, k is the Coulomb constant, Q1 and Q2 are the magnitudes of the charges, and r is the distance between the charges.

2. How do I determine the direction of the total Coulomb force vector?

The direction of the total Coulomb force vector is determined by the direction of the individual forces between the charges. If the charges are of the same sign, the forces will repel each other and the total force will be in the opposite direction. If the charges are of opposite signs, the forces will attract each other and the total force will be in the same direction.

3. What are the units of the total Coulomb force vector?

The units of the total Coulomb force vector are Newtons (N), which is equivalent to kg*m/s^2. This is the standard unit of force in the International System of Units (SI).

4. Can the total Coulomb force vector be negative?

Yes, the total Coulomb force vector can be negative. This occurs when the charges have opposite signs and the forces are directed in opposite directions. In this case, the negative sign indicates that the forces are attractive rather than repulsive.

5. How does the distance between the charges affect the total Coulomb force vector?

The total Coulomb force vector is inversely proportional to the square of the distance between the charges. This means that as the distance increases, the force decreases. Similarly, as the distance decreases, the force increases. This relationship is known as the inverse square law.

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