Consider an infinitely long and thin line of charge, with density 8nC/m. Calculate the electric potential difference ((phi)1-2) between two points in air at radial disances 1mm and 3mm.
I am assuming:
Phi(r) = λ/2(pi)ε0 * ln(rR/r) where rR is a reference point
The Attempt at a Solution
Our lecturer told us that we did not have to derive electric potential - just use what we had discussed in the notes, which, as far I can tell, is the above.
As far as I can see, this is just plugging in 8x10^-9 c for λ, 3x10^-3m for rR, and 1x10^-3m for r.
The problem: this is a 20 point question! One fifth of the assignment's points.
I feel like there is something to this that I am missing, but, despite having poured through our notes and researched on the internet (and even found a worked problem that is similar), I cannot find anything else to do other than plug in for λ, rR, and r.
If I'm wrong and there's a lot more I have to do, could someone please help point me in the right direction?
The potential at position r due to a point charge q at position r' is
phi(r) = q/4piε0 * 1/|r-r'|
a.) Calculate grad phi and hence the electric field E.
b.) What is the force experienced by a charge q1 at position r?
c.) What is the potential energy of the charge at q1?
Given above; however, also worth knowing is
E = -grad phi
U = q2 * phi
The Attempt at a Solution
This is like the above question, where, in my mind at least, the points allocated don't seem to match up with the amount of work to be done.
Part a I have no problems with. I thought of r in terms of x,y, and z (again, according to our lecturer's advice), and then I used partial derivative with respect to x, then stated that, according to symmetry, y and z worked out to be the same, giving an answer of
grad phi = q/4piε0 * r'-r/|r-r'|^3
and since E = -grad phi, this reversed the top value (r'-r) to give
E = = q/4piε0 * r-r'/|r-r'|^3.
That felt like a solid 10 points worth of work.
To do part b, it seems that all I have to do is multiply another q into the equation, and multiply that by -1 (since the charge is going from q1 (which I assume is equivalent to "q2") to q (which I assume is equivalent to "q1")). This feels more like 2 or 3 points worth of work.
Then for part c, I have done a negative integral of what I calculated for F, with respect to r. This gives me
U = q*q1/4piε0 * 1/|r-r'|
Not only does this answer not seem correct, but it also doesn't feel like 10 points worth of work.
Again, if someone could help me see where I went wrong, and point me in the direction of whatever I'm missing, I would be very grateful.