An infinitely long, uniformly charged straight line has linear charge density ?1
ID: 2283119 • Letter: A
Question
An infinitely long, uniformly charged straight line has linear charge density ?1 coul/m. A straight rod of length 'b' lies in the plane of the straight line and perpendicular to it, with its enared end at distance 'a' from the line. The charge density on the rod varies with distance 'y', measured from the lower end, according to ?(on rod) = (?2*b)/(y+a), where ?2 is a constant. Find the electrical force exerted on the rod by the charge on the infinite straight line, in the ?1, ?2, a, and b, and constants like ?0.
Explanation / Answer
Relevant equations Gauss's Law.
I first treated the problem as if there was only a point P above the infinite line and applied Gauss's Law using a cylinder as my Gaussian surface. My answer was E = (?2 * b) / [2pi*?0*(a+b)^2]
It's correct to use Gauss law to find the field of the line, but I think that you made a mistake because your answer is independent of ?1. Also your answer is not general, because point P is not a random point, it is rod's upper end (its distance from the line is a+b). Now let's see a way to solve the problem. Firstly use Gauss law to find the field E(r) of the line at a random point which distance from the line is a variable r. This r must be the radius of your Gaussian cylinder. Then take an infinitesimally small part of the rod, with length dy, which distance from the lower end of the rod is y (and consequently from the line is r=y+a). This part is charged with charge dq, so using E(r) we can compute the force dF exerted on it. Note that you can also find dq in terms of y and dy using rod's charge density. Finally you get an equation for dF in terms of y, dy and some constants.