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Chapter 4: Q4.30P (page 205)

An electric dipole p鈫�, pointing in the ydirection, is placed midwaybetween two large conducting plates, as shown in Fig. 4.33. Each plate makes a small angle 胃with respect to the xaxis, and they are maintained at potentials 卤V.What is the directionof the net force onp鈫�?(There's nothing to calculate,here, butdo explain your answer qualitatively.)

Short Answer

Expert verified

The net force onp鈫�placed midway between two large conducting plates is towards the right.

Step by step solution

01

Given data

There is a dipole with dipole moment p鈫�pointing in the ydirection and placed midway

between two large conducting plates.

The plate makes asmall angle 胃with respect to the xaxis.

The plates maintained at potentials 卤V.

02

Determine the direction of force on a dipole

The lines of force and the corresponding electric field are always perpendicular to a conducting surface.Thus, for the setup mentioned in the problem, the lines of follows emerge from the upper plate kept at a positive potential and go into the lower plate kept at a negative potential as follows

The corresponding direction of forces on the positive and negative ends of the dipole are also shown in the above figure.

Thus, the net force is towards the right.

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Most popular questions from this chapter

A conducting sphere at potential V0 is half embedded in linear dielectric material of susceptibility 蠂e, which occupies the regionz<0 (Fig. 4.35).

Claim:the potential everywhere is exactly the same as it would have been in the

absence of the dielectric! Check this claim, as follows:

  1. Write down the formula for the proposed potentialrole="math" localid="1657604498573" V(r),in terms ofV0,R,andr.Use it to determine the field, the polarization, the bound charge, and the free charge distribution on the sphere.
  2. Show that the resulting charge configuration would indeed produce the potentialV(r).
  3. Appeal to the uniqueness theorem in Prob. 4.38 to complete the argument.
  4. Could you solve the configurations in Fig. 4.36 with the same potential? If not, explain why.

E2鈫�Find the field inside a sphere of linear dielectric material in an otherwise uniform electric field E0鈫�(Ex. 4.7) by the following method of successive approximations: First pretend the field inside is just E0鈫�, and use Eq. 4.30 to write down the resulting polarization P0鈫�. This polarization generates a field of its own, E1鈫� (Ex. 4.2), which in turn modifies the polarization by an amount P1鈫�. which further changes the field by an amount E2鈫�, and so on. The resulting field is E鈫�0+E鈫�1+E鈫�2+.... . Sum the series, and compare your answer with Eq. 4.49.

Earnshaw's theorem (Prob. 3.2) says that you cannot trap a charged

particle in an electrostatic field. Question:Could you trap a neutral (but polarizable) atom in an electrostatic field?

(a) Show that the force on the atom is F鈫�=12伪鈭�鈫�E2

(b) The question becomes, therefore: Is it possible for E2 to have a local maximum (in a charge-free region)? In that case the force would push the atom back to its equilibrium position. Show that the answer is no. [Hint:Use Prob. 3.4(a).]

According to Eq. 4.5, the force on a single dipole is (p 路 V)E, so the

netforce on a dielectric object is

F=鈭�P路鈭�Eextd蟿

[Here Eextis the field of everything except the dielectric. You might assume that it wouldn't matter if you used the total field; after all, the dielectric can't exert a force on itself. However, because the field of the dielectric is discontinuous at the location of any bound surface charge, the derivative introduces a spurious delta function, and it is safest to stick withEext Use Eq. 4.69 to determine the force on a tiny sphere, of radius , composed of linear dielectric material of susceptibility 蠂ewhich is situated a distance from a fine wire carrying a uniform line charge位 .

A spherical conductor, of radius a,carries a charge Q(Fig. 4.29). It

is surrounded by linear dielectric material of susceptibilityXeout to radius b.Find the energy of this configuration (Eq. 4.58).
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