Find the electric field a distance z above the center of a square loop (side a) carrying...

Find the electric field a distance z above the center of a square loop (side a) carrying uniform line charge λ (Fig. 2.8). [Hint: Use the result of Ex. 2.2.]

Reference: Fig. 2.8.

Reference: Ex. 2.2.

Find the electric field a distance z above the midpoint of a straight line segment of length 2L that carries a uniform line charge λ (Fig. 2.6).

Solution: The simplest method is to chop the line into symmetrically placed pairs (at ±x), quote the result of Ex. 2.1 (with d/2 → x, q → λ dx), and integrate (x : 0 → L).But here’s a more general approach:

For points far from the line (z >>L),

This makes sense: From far away the line looks like a point charge q = 2λL. In the limit L →∞, on the other hand, we obtain the field of an infinite straight wire: