(Electric field induced by two charges)Given a positive chargeQand a negative charge−Q,a d...

(Electric field induced by two charges)Given a positive chargeQand a negative charge−Q,a distance 2a apart, let us introduce a coordinate system as in the figure below. Since the charges lie on the z axis, it follows that the electric field that they induce will be symmetric about thezaxis.

With the pointPfixed, imagine lettingabecome arbitrarily small. Show, from (12.2), that we obtain

asa→ 0. Thus, Φ(ρ, Ø) 0 as a→ 0, as makes sense, because the positive and negative charges cancel each other as they are moved together. However, observe that if, asais decreased,Qis increased such that the productQais held constant, then (12.3) becomes

whereμ= 2Qais called the dipole moment, and the charge configuration is said to constitute an electric dipole. If, for instance, a molecule is comprised of equal and opposite charges,+Qand−Q,displaced by a very small distance 2a, then, even thoughais not tending to zero andQto infinity, the field induced by that molecule is approximately equal to that of an idealized dipole of strength μ =2Qa,at points sufficiently far away (i.e., forρ/a≫ 1).