Suppose the field inside a large piece of magnetic material is B0, so that H0=(1/μ0)B0-M, where M is a "frozen-in" magnetization.

(a) Now a small spherical cavity is hollowed out of the material (Fig. 6.21). Find the field at the center of the cavity, in terms of B0 and M. Also find H at the center of the cavity, in terms of H0 and M.

(b) Do the same for a long needle-shaped cavity running parallel to M.

(c) Do the same for a thin wafer-shaped cavity perpendicular to M.

Figure 6.21

Assume the cavities are small enough so M, B0, and H0 are essentially constant. Compare Prob. 4.16. [Hint: Carving out a cavity is the same as superimposing an object of the same shape but opposite magnetization.]

Short Answer

Expert verified

(a)

The values of field at the center of the cavity, in terms of B0 andis B=B0-23μ0M.

The value of at the center of the cavity, in terms of H0 and is H=H0+M3.

(b)

The value of field for long needle-shaped cavity perpendicular to M is B=B0-μ0M.

The value of for long needle-shaped cavity perpendicular to M is H=H0.

(c) The value of for a thin wafer-shaped cavity perpendicular to M is H=1μ0.

Step by step solution

01

Write the given data from the question.

Suppose the field inside a large piece of magnetic material is B0.

Consider a small spherical cavity is hollowed out of the material in (Fig. 6.21).

Assume the cavities are small enough so M, B0, and H0 are essentially constant.

02

Determine the formula of field at the center of the cavity, H at the center of the cavity, field for long needle-shaped cavity perpendicular to M and H for long needle-shaped cavity perpendicular to M.

Write the formula offield at the center of the cavity, in terms of .B0

B=B0+Bs …… (1)

Here, B0is field inside a large piece of magnetic material and Bsis field inside of a uniformly magnetized sphere.

Write the formula of at the center of the cavity, in terms of H0 and M

H=1μ0B …… (2)

Here, μ0is permeability and B is magnetic field at the center of the cavity in terms of B0.

Write the formula offield for long needle-shaped cavity perpendicular to M.

B=B0+Bs …… (3)

Here, role="math" localid="1657695281686" B0is field inside a large piece of magnetic material and Bsis field inside of a uniformly magnetized sphere.

Write the formula of H for long needle-shaped cavity perpendicular to M.

role="math" localid="1657695633493" H=1μ0B …… (4)

Here, μ0is permeability and B is magnetic field at the center of the cavity in terms of B0.

Write the formula of H for a thin wafer-shaped cavity perpendicular to M.

H=1μ0B …… (5)

Here, μ0is permeability and B is magnetic field at the center of the cavity in terms of B0.

03

(a) Determine the value of field at the center of the cavity, in terms of B0 and M.

The field inside of a uniformly magnetized sphere with magnetization -Mis:

role="math" localid="1657696622237" Bs=-23μ0M

Determine the field at the center of the cavity, in terms of B0 and.

Substitute -23μ0Mfor Bsinto equation (1).

role="math" localid="1657696633523" B=B0-23μ0M

Therefore, the values of field at the center of the cavity, in terms of B0 and M is B=B0-23μ0M.

Determine the H at the center of the cavity, in terms of H0 and M.

Substitute B0for Binto equation (2).

H=1μ0B0-23M=1μ0μ0M+μ0H0-23M=H0+M3

Therefore, the value of at the center of the cavity, in terms of H0 and M is H=H0+M3.

04

(b) Determine the value of value of field for long needle-shaped cavity perpendicular to M and H for long needle-shaped cavity perpendicular to M.

The needle cavity's induced field will resemble an endless cylinder with magnetization -Min this scenario. It causes a field that is:

role="math" localid="1657697322029" Bs=-μ0M

Determine the field for long needle-shaped cavity perpendicular to M.

Substitute -μ0Mfor Bsinto equation (3).

B=B0-μ0M

Determine the H for long needle-shaped cavity perpendicular to M.

Substitute B0-μ0Mfor Binto equation (4).

H=1μ0B0-μ0M=1μ0B0-M=H0

05

(c) Determine the value of field for a thin wafer-shaped cavity and H for a thin wafer-shaped cavity perpendicular to M.

The cavity will only generate a little magnetic field if the wafer is very thin, and the field will be:

Determine the field for a thin wafer-shaped cavity.

B=B0

Determine the H for a thin wafer-shaped cavity perpendicular to M.

H=H0+M

Therefore, the value of H for a thin wafer-shaped cavity perpendicular to M is H=1μ0.

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Most popular questions from this chapter

On the basis of the naïve model presented in Sect. 6.1.3, estimate the magnetic susceptibility of a diamagnetic metal such as copper. Compare your answer with the empirical value in Table 6.1, and comment on any discrepancy.

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?

Notice the following parallel:

{·D=0×E=0,ε0E=D-P(Nofreecharge)·B=0×H=0,μ0H=B-μ0M(Nofreecharge)

Thus, the transcription DB,EH,Pμ0M,ε0μ0,, turns an electrostatic problem into an analogous magnetostatic one. Use this, together with your knowledge of the electrostatic results, to rederive.

(a) the magnetic field inside a uniformly magnetized sphere (Eq. 6.16);

(b) the magnetic field inside a sphere of linear magnetic material in an otherwise uniform magnetic field (Prob. 6.18);

(c) the average magnetic field over a sphere, due to steady currents within the sphere (Eq. 5.93).

A short circular cylinder of radius and length L carries a "frozen-in" uniform magnetization M parallel to its axis. Find the bound current, and sketch the magnetic field of the cylinder. (Make three sketches: one forL>>a, one forL<<a, and one forLa.) Compare this bar magnet with the bar electret of Prob. 4.11.

An iron rod of length Land square cross section (side a) is given a uniform longitudinal magnetization M, and then bent around into a circle with a narrow gap (width w), as shown in Fig. 6.14. Find the magnetic field at the center of the gap, assuming waL.

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