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Introduction to Electrodynamics

David J. Griffiths

Physics

4th edition Edition · 613 pages

ISBN: 9780321856562

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 $\vec{D}$ , 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

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

^{, $B (r) ≅ B (r_{0}) + [(r - r_{0}) \cdot \nabla_{0}] B (r_{0})$}

^{Where $r_{0}$ the position of the dipole and $\nabla_{0}$ is denotes differentiation with respect to $r_{0}$ . Put this into the Lorentz force law (Eq. 5.16) to obtain}

^{. $F = I \oint d I \times [(r \cdot \nabla_{0}) B (r_{0})]$}

^{Or, numbering the Cartesian coordinates from 1 to 3:}

^{$F_{i} = I \sum_{j, k, l = 1}^{3} ε_{i j k} {\oint r_{l} d l_{j}} [\nabla_{0 l} B_{k} (r_{0})]$ ,}

^{Where $ε_{ijk}$ is the Levi-Civita symbol ( $+ 1$ if $ijk = 123, 231$ , or $312$ ; $- 1$ if $ijk = 132$ , $213$ , or $321; 0$ otherwise), in terms of which the cross-product can be written ${(A \times B)}_{i} = \sum_{j, k = 1}^{3} ε_{ijk} A_{j} B_{k}$ . Use Eq. 1.108 to evaluate the integral. Note that}

^{$\sum_{j = 1}^{3} ε_{ijk} ε_{ljm} = δ_{il} δ_{km} - δ_{im} δ_{kl}$}

^{Whereoil is the Kronecker delta (Prob. 3.52).}

A current $I$ flows down a long straight wire of radius. If the wire is made of linear material (copper, say, or aluminium) with susceptibility $X_{m}$ , and the current is distributed uniformly, what is the magnetic field a distance $s$ from the axis? Find all the bound currents. What is the net bound current flowing down the wire?

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?

lf $J_{f} = 0$ everywhere, the curl of $H$ vanishes (Eq. 6.19), and we can express $H$ as the gradient of a scalar potential W:

$H = - \nabla W$

According to Eq. 6.23, then,

$\nabla^{2} W = (\nabla \cdot M)$

So $W$ obeys Poisson's equation, with $\nabla \cdot M$ as the "source." This opens up all the machinery of Chapter 3. As an example, find the field inside a uniformly magnetized sphere (Ex. 6.1) by separation of variables.

A familiar toy consists of donut-shaped permanent magnets (magnetization parallel to the axis), which slide frictionlessly on a vertical rod (Fig. 6.31). Treat the magnets as dipoles, with mass m_d and dipole moment m.

(a) If you put two back-to-hack magnets on the rod, the upper one will "float"-the magnetic force upward balancing the gravitational force downward. At what height (z) does it float?

(b) If you now add a third magnet (parallel to the bottom one), what is the ratio of the two heights? (Determine the actual number, to three significant digits.) $[A n s w e r : (a) 3 μ_{0} m^{2} (b) 0.8501]$

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