In this problem we will calculate the electric field at some distance from the midpoint of a charged wire. Take the wire of length 2L to lie on the x axis from x = -L to x = +L. The wire has charge per unit length (so, for example, the total charge on the wire is 2L). Let's determine the field at a point P at x=y=0 and third coordinate given by z

Start by examining the field due to a infinitesimal piece of wire of length dx.

a. In terms of the variables in this problem, how much charge is on this bit of wire of length dx?

b. If this element of the wire is at position x, how far is it from the point P?

c. Find an expression for the magnitude of the infinitesimal electric field dE at P due to this bit of wire.

d. Find expressions for the x, y, and z components of dE ( dEx, dEy, and dEz ) at P. One of these components is immediately seen to be zero. Which one?

e. Find an expression for the total values of the remaining two components of the field E (Ex, Ey, and/or Ez) as integrals over the full length of the wire of your expressions for dEx, dEy, and/or dEz

f. One additional component can be seen to be equal to zero, either by symmetry or from the integral expression. Which one is that? Why?

g. The remaining integral is in the table of integrals in the back of your book. Carry it out, and find the expression for the electric field at P.

h. Show that for a wire much longer than the distance z to the point, your expression reduces to E = (/2₀)(1/z).