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Chapter 4: Q4.7P (page 172)

Show that the energy of an ideal dipole p in an electric field E isgiven by

U=鈭�p鈰�E鈭�

Short Answer

Expert verified

Thepotential energy for the dipole moment is鈭�p鈰�E .

Step by step solution

01

Determine the formulas

Consider the formula for the torque on the dipole moment as

蟿=辫贰蝉颈苍胃

Here, pis the dipole moment and E is the electric field.

02

Determine the formula for the energy of the dipole moment 

Consider the formula for work done in the rotating dipole moment as:

dw=蟿d胃

Substitute the values and solve as

dw=pEsin胃d胃w=鈭�a1a2pEsin胃d胃w=pE(cos胃1鈭�cos胃2)

Consider the change in the potential for the two dipole positions and the corresponding working done as

w=U(胃2)鈭�U(胃1)=鈭�pE(cos胃2鈭�胃1)=鈭�pEcos胃=鈭�p鈰�E

Therefore, the potential energy for the dipole moment is 鈭�p鈰�E.

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Most popular questions from this chapter

According to Eq. 4.1, the induced dipole moment of an atom is proportional to the external field. This is a "rule of thumb," not a fundamental law,

and it is easy to concoct exceptions-in theory. Suppose, for example, the charge

density of the electron cloud were proportional to the distance from the center, out to a radius R.To what power of Ewould pbe proportional in that case? Find the condition on such that Eq. 4.1 will hold in the weak-field limit.

Calculate the potential of a uniformly polarized sphere (Ex. 4.2) directly from Eq. 4.9.

A thick spherical shell (inner radius a, outer radius b) is made of dielectric material with a "frozen-in" polarization

P(r)=krr^

Where a constant and is the distance from the center (Fig. 4.18). (There is no free charge in the problem.) Find the electric field in all three regions by two different methods:

Figure 4.18

(a) Locate all the bound charge, and use Gauss's law (Eq. 2.13) to calculate the field it produces.

(b) Use Eq. 4.23 to find D, and then getE from Eq. 4.21. [Notice that the second method is much faster, and it avoids any explicit reference to the bound charges.]

Suppose the region abovethe xyplane in Ex. 4.8 is alsofilled withlinear dielectric but of a different susceptibility 蠂'e.Find the potential everywhere.

(a) For the configuration in Prob. 4.5, calculate the forceon p鈫�2due to p鈫�1and the force on p鈫�1due to p鈫�2. Are the answers consistent with Newton's third law?

(b) Find the total torque on p鈫�2 with respect to the center ofp鈫�1and compare it with

the torque onp鈫�1 about that same point. [Hint:combine your answer to (a) with

the result of Prob. 4.5.]
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