Question

Part A

Consider two identical dipoles, each consisting of charges +q and ?q separated by a distance d and oriented as shown in the figure (Figure 1)a. Calculate the electric potential energy, expressed in terms of the electric dipole moment p=qd, for the situation where r?d. Ignore the potential energy involved in forming each individual molecule

Express your answer in terms of the variables p, r, and appropriate constants.

Part C

Repeat part A for the orientation of the dipoles shown in the figure b. The dipole interaction is more complicated when we have to average over the relative orientations of the two dipoles due to thermal motion or when the dipoles are induced rather than permanent.

Express your answer in terms of the variables p, r, and appropriate constants.

Answer 1

The identical dipoles consists of charges +q and -q. The distance between the charges is d. Electric potential energy is expressed by electric dipole moment p=dq for the situation where r>>d.

An electric dipole is formed by two point charges +q and ?q connected by a vector d. The electric dipole moment is defined by convention the vector d points from the negative to the positive charge. Here we also take the origin to be at the center and d to be aligned to the z axis.

We first calculate the potential and then the field:

V = 1 / (4_o) * (q / r)

Where r (positive / negative) are the distances from the positive and negative charge to the point r.

Now we first calculate the potential and then the field:

Where r (positive / negative) are the distances from the positive and negative charge to the point r.

Now

r² + /- = (r² + d. r + d² / 4)

           = r2 (1 +/- d/r cos + d² / 4 r²⁾

Now consider the “far field limit” r>>d.

1 / r + =   1 / r (1 + d/r cos + d² / 4 r²) ^-1/2

          = 1 / r (1 + d/ 2 r cos + O ((d / 4 r) ²))

The far-field limit is

V = q d cos/ 4 ₀r² =    p · r / 4 ₀r²       

The components of the electric field E= ??V are simplest in spherical polar coordinates:

               E_r = =      2 p cos/ 4 _o r³  

               E_o= - 1 / r ( ) = p sin/ 4_o r³   

     To get a co-ordinate free form of the electric field we can use

E = - V = - (1 / 4 o) ( (p. / r²⁾)