Question

A rod of length 2l lies on the x-axis from x = −l to x = +l. The left half of the rod carries uniform negative charge density −λ while the right half carries a uniform positive charge density +λ. (a) What is the net charge of each half of the rod? What is the total charge of the entire rod? (b) Determine an exact expression for the magnitude of the electric field at an arbitrary point along the x-axis a distance r from the center of the rod. Use Worked Problem 23.5 as a guide. Combine multiple terms into a single fraction. (c) Approximate your expression for r ≫ l. You should find E ≈ 2kλl2 r 3 . (d) The charge distribution on this rod resembles a dipole, only instead of point charges ±q, the charge is spread out. Determine the dipole moment of this distribution. To do this, first note that ~p should point from negative to positive, thus in the ˆı direction. Next, compare the approximate field expression above to that of a point-charge dipole in the same configuration (at a distance r away along the dipole axis). (e) Show that you can interpret this result as a pair of point charges ±q separated by a distance l. Does that make physical sense here?

Answer 1