Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information....

Field of a uniformly charged rod


A rod of length L has charge Q spread uniformly along it. Find the electric field at a point of distance d from the center of the rod, along the rod's axis.

Solution: Let x=0 be the center of the rod, and let the positive x axis be to the right. This is a one-dimensional situation, so we really only need to do a single integral representing the total field along the x axis. We imagine breaking the rod down into short pieces of length dx, each with charge dq. Since charge is uniformly spread along the rod, we have dq=(dx/L)Q. Since the pieces are infinitesimally short, we can treat them as point charges and use the expression kdq/r2 for their contributions to the field, where r=d-x is the distance from the charge at x to the point in which we are interested.

The integral can be looked up in a table, or reduced to an elementary form by substituting a new variable for d-x. The result is

For large values of d, the expression in brackets gets smaller for two reasons: (1) the denominators of the fractions become large, and (2) the two fractions become nearly the same, and tend to cancel out. This makes sense, since the field should get weaker for larger values of d.

It is also interesting to note that the field becomes infinite at the ends of the rod, but is not infinite on the interior of the rod. Can you explain physically why this happens?

Last Update: 2009-07-21