SERWAY 23.20
A continuous line of charge lies along the x axis, extending from the point x = +x0 out to
x equals positive infinity. The line carries charge with a uniform linear charge density of
λ0. What are the magnitude and direction of the electric field at the origin?
Let’s evaluate the direction of the E-field at the origin first. Think of the line of charge as
a collection of little positive point charges, evenly spaced along the x axis. If you’re
standing at the origin, each of those little point charges will generate an E-field that
points radially away. Thus, at the origin, the E-field will have no y component, and will
point to-the-left, in the negative y direction.
Finding the magnitude is a little harder…
dE  k e
dq
dq
 ke 2
2
r
x
But, dq  0 dx .
Thus:
dE  k e
0 dx
x2

dx
2
x0 x
E   dE  k e 0 
 k e 0
E
x
E

x0
1 1  k 
  k e 0     e 0
x0
  x0 
k e 0
, and points to-the-left (assuming a positive charge distribution).
x0