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

Short Answer

Expert verified

The net force onpplaced midway between two large conducting plates is towards the right.

Step by step solution

01

Given data

There is a dipole with dipole moment ppointing in the ydirection and placed midway

between two large conducting plates.

The plate makes asmall angle θwith respect to the xaxis.

The plates maintained at potentials ±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

E2Find the field inside a sphere of linear dielectric material in an otherwise uniform electric field E0(Ex. 4.7) by the following method of successive approximations: First pretend the field inside is just E0, and use Eq. 4.30 to write down the resulting polarization P0. This polarization generates a field of its own, E1 (Ex. 4.2), which in turn modifies the polarization by an amount P1. which further changes the field by an amount E2, and so on. The resulting field is E0+E1+E2+.... . Sum the series, and compare your answer with Eq. 4.49.

The Clausius-Mossotti equation (Prob. 4.41) tells you how to calculatethe susceptibility of a nonpolar substance, in terms of the atomic polariz-ability. The Langevin equation tells you how to calculate the susceptibility of apolar substance, in terms of the permanent molecular dipole moment p. Here's howit goes:

(a) The energy of a dipole in an external field E isu=-p·Ecosθ

(Eq. 4.6), whereθ is the usual polar angle, if we orient the z axis along E.

Statistical mechanics says that for a material in equilibrium at absolute temperature

T, the probability of a given molecule having energy u is proportional to

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exp(-u/kT)

The average energy of the dipoles is therefore

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where =sinθdθdϕ, and the integration is over all orientations θ:0π;ϕ:02πUse this to show that the polarization of a substance

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P=Np[cothpE/kT-kT/pE] (4.73)

That's the Langevin formula. Sketch as a function ofPE/KT .

(b) Notice that for large fields/low temperatures, virtually all the molecules arelined up, and the material is nonlinear. Ordinarily, however, kT is much greaterthan p E. Show that in this regime the material is linear, and calculate its susceptibility,in terms of N, p, T, and k. Compute the susceptibility of water at 20°C,and compare the experimental value in Table 4.2. (The dipole moment of wateris 6.1×10-30C·m) This is rather far off, because we have again neglected thedistinction between E and Eelse· The agreement is better in low-density gases,for which the difference between E and Eelse is negligible. Try it for water vapor

at 100°C and 1 atm.

A conducting sphere of radius a, at potential V0, is surrounded by a

thin concentric spherical shell of radius b,over which someone has glued a surface charge

σ(θ)=kcosθ,

where k is a constant and θis the usual spherical coordinate.

a) Find the potential in each region: (i) r>b, and (ii) a<r<b.

b) Find the induced surface charge σi(θ)on the conductor.

c) What is the total charge of this system? Check that your answer is consistent with the behavior of V at large.

A conducting sphere at potential V0 is half embedded in linear dielectric material of susceptibility χe, 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.

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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