Download Electric Dipoles: Properties and Interactions and more Study notes Chemistry in PDF only on Docsity! EM 3 Section 5: Electric Dipoles An electric dipole is formed by two point charges +q and −q connected by a vector a. The electric dipole moment is defined as p = qa . By convention the vector a points from the negative to the positive charge. Here we also take the origin to be at the centre and a to be aligned to the z axis (see diagram) Figure 1: Diagram of electric dipole aligned along z axis. Griffiths Fig 3.37 5. 1. Field of an electric dipole We first calculate the potential and then the field: V = q 4πε0 ( 1 r+ − 1 r− ) (1) where r± are the distances from the +ve(-ve) charge to the point r. Now r2 ∓ = |r ± a/2|2 = (r2 ± a · r + a2/4) = r2 ( 1± a r cos θ + a2 4r2 ) Now consider the “far field limit” r a 1 r± = 1 r ( 1∓ a r cos θ + a2 4r2 )−1/2 ' 1 r ( 1± a 2r cos θ +O((a/r)2) ) where O((a/r)2) indicates terms proportional to (a/r)2 or higher powers Thus we obtain in the far-field limit V = qa cos θ 4πε0r2 = p · r̂ 4πε0r2 (2) One can check that away from the charges this is a solution of Laplace’s equation (see tut) 1 The components of the electric field E= −∇V are simplest in spherical polar coordinates: Er = −∂V ∂r = 2p cos θ 4πε0r3 Eθ = −1 r ∂V ∂θ = p sin θ 4πε0r3 (3) To get a co-ordinate free form of the electric field we can use (see tutorial sheet 1) E = −∇V = − 1 4πε0 ∇ ( p · r̂ r2 ) = 1 4πε0 ( 3(p · r)r r5 − p r3 ) (4) The important point to note is that a dipole field is 1/r3, whereas a point charge field is 1/r2 N.B. The above ‘far-field’ limit r a can also be presented as an ideal dipole which is the limit a→ 0, q →∞ but p finite. The ideal dipole is a useful approximation to the ‘physical dipole’ for which the potential is given by (1) and E = −∇V = q 4πε0 [ r − a/2 |r − a/2|3 − r + a/2 |r + a/2|3 ] (5) However the sketches look a little different: Figure 2: Sketch of electric dipole field: ideal and ‘physical’ Griffiths Fig 3.37 Why dipoles matter I: Many molecules have a permanent dipole moment p (e.g. H20) All others, and all atoms, acquire an induced dipole when placed in E field Since atoms and molecules are (a) neutral and (b) almost pointlike, the ideal dipole concept is crucial to understanding media see later. 5. 2. Interaction of dipole with external Electric Field To calculate the force on a dipole in an external field Eext = −∇φ (note here we use φ for the electrostatic potential of the external field) it is simplest to first calculate the potential energy of the dipole in this field: Udip = qφ(a/2)− qφ(−a/2) ' qa · ∇φ = −p · Eext where we have made a Taylor expansion to first order in a. 2
