Calculating the work done in an electric field

[Biker]

Homework Statement

It is not an actual questions, Just misconceptions
For example,
Lets find the work done over a distance in an electric field

Homework Equations

v = PE/q
E = F/q
F = k q1 q2 /r^2
W = fd

The Attempt at a Solution

I didn't study calculus yet but I am going to because I need to get to know these equations.

I know that I can't use the formula w = f d because obviously the f varies if I don't have a uniform EF
these equations only take place when the electric field is constant otherwise I need to use calculus to find the equation because the forces varies with distance.

So in V = PE/C
I can't just substitute PE with F d... so if you are able to put the proof of V = kq/r (If it is related to calculus) that would be great so I can check it when I am done with calculus.

 

[BvU]

Perhaps you already know a bit about derivatives ? If you know that $${d\over dx} {1\over x} = -{1\over x^2}$$you can see that that kind of matches ## V = {kQ\over r} ## and ##E = - k {Q\over r^2 }##.

Indeed $$V(r_2) - V(r_1) = \int_{r_1}^{r_2} F(r)\, dr = -k Q \int_{r_1}^{r_2} {1\over r^2} \, dr = -kQ \left [- {1\over r} \right ]_{r_1}^{r_2} = k{Q \over r_2} - k{Q\over r_1}$$