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Chapter 6: Q14P (page 284)

For the bar magnet of Problem. 6.9, make careful sketches of M, B, and H, assuming L is about 2a. Compare Problem. 4.17.

Short Answer

Expert verified

Draw the careful sketches of for the bar magnet.

Draw the careful sketches of for the bar magnet.

Draw the careful sketches of for the bar magnet.

Step by step solution

01

Write the given data from the question.

Reference as problem 6.9.

Assuming is about 2a.

02

Draw careful sketches of M, B, and H.

Draw the circuit diagram of M for the bar magnet.

Figure 1

Draw the circuit diagram of for the bar magnet.

Figure 2

Draw the circuit diagram of for the bar magnet.

Figure 3

We observe that the polarisation and the magnetization fields are similar (in problem 4.17).

Similar to the electric field, the auxiliary field H has a discontinuity at the top and bottom of the cylinder.

Last but not least, the magnetic field resembles the displacement field D→, because the field lines in the electric case loop back on themselves because there are no free charges (like the magnetic field).

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Most popular questions from this chapter

A coaxial cable consists of two very long cylindrical tubes, separated by linear insulating material of magnetic susceptibility χm. A currentI flows down the inner conductor and returns along the outer one; in each case, the current distributes itself uniformly over the surface (Fig. 6.24). Find the magnetic field in the region between the tubes. As a check, calculate the magnetization and the bound currents, and confirm that (together, of course, with the free currents) they generate the correct field.

Figure 6.24

Calculate the torque exerted on the square loop shown in Fig. 6.6, due to the circular loop (assume is much larger than or ). If the square loop is free to rotate, what will its equilibrium orientation be?

Imagine two charged magnetic dipoles (charge q, dipole moment m), constrained to move on the z axis (same as Problem 6.23(a), but without gravity). Electrically they repel, but magnetically (if both m's point in the z direction) they attract.

(a) Find the equilibrium separation distance.

(b) What is the equilibrium separation for two electrons in this orientation. [Answer: 4.72x10-13m.]

(c) Does there exist, then, a stable bound state of two electrons?

A current Iflows down a long straight wire of radius. If the wire is made of linear material (copper, say, or aluminium) with susceptibility Xm, and the current is distributed uniformly, what is the magnetic field a distances from the axis? Find all the bound currents. What is the net bound current flowing down the wire?


In Prob. 6.4, you calculated the force on a dipole by "brute force." Here's a more elegant approach. First writeB(r)as a Taylor expansion about the center of the loop:

,B(r)≅B(r0)+[(r−r0)⋅∇0]B(r0)

Wherer0the position of the dipole and ∇0is denotes differentiation with respect tor0. Put this into the Lorentz force law (Eq. 5.16) to obtain

.F=I∮dI×[(r⋅∇0)B(r0)]

Or, numbering the Cartesian coordinates from 1 to 3:

Fi=I∑j,k,l=13εijk{∮rldlj}[∇0lBk(r0)],

Where εijk is the Levi-Civita symbol (+1ifijk=123,231, or312; −1ifijk=132, 213, or 321;0otherwise), in terms of which the cross-product can be written (A×B)i=∑j,k=13εijkAjBk. Use Eq. 1.108 to evaluate the integral. Note that

∑j=13εijkεljm=δilδkm−δimδkl

Whereoil is the Kronecker delta (Prob. 3.52).
See all solutions

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