Holooly Plus Logo
Holooly Plus Logo

Question 5.60: A uniformly charged solid sphere of radius R carries a total...

A uniformly charged solid sphere of radius R carries a total charge Q, and is set spinning with angular velocity ω about the z axis.

(a) What is the magnetic dipole moment of the sphere?

(b) Find the average magnetic field within the sphere (see Prob. 5.59).

(c) Find the approximate vector potential at a point (r, θ) where r \gg R .

(d) Find the exact potential at a point (r, θ) outside the sphere, and check that it is consistent with (c). [Hint: refer to Ex. 5.11.]

(e) Find the magnetic field at a point (r, θ) inside the sphere (Prob. 5.30), and check that it is consistent with (b).

The Blue Check Mark means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.
Learn more on how we answer questions.

(a) Problem 5.53 gives the dipole moment of a shell: m =\frac{4 \pi}{3} \sigma \omega R^{4} \hat{ z } . \text { Let } R \rightarrow r, \sigma \rightarrow \rho d r and integrate:

m =\frac{4 \pi}{3} \omega \rho \hat{ z } \int_{0}^{R} r^{4} d r=\frac{4 \pi}{3} \omega \rho \frac{R^{5}}{5} \hat{ z } . \quad \text { But } \rho=\frac{Q}{(4 / 3) \pi R^{3}}, \text { so } m =\frac{1}{5} Q \omega R^{2} \hat{ z } .

(b)  B _{ ave }=\frac{\mu_{0}}{4 \pi} \frac{2 m }{R^{3}}=\frac{\mu_{0}}{4 \pi} \frac{2 Q \omega}{5 R} \hat{ z } .

(c)  A \cong \frac{\mu_{0}}{4 \pi} \frac{m \sin \theta}{r^{2}} \hat{\phi}=\frac{\mu_{0}}{4 \pi} \frac{Q \omega R^{2}}{5} \frac{\sin \theta}{r^{2}} \hat{\phi} .

(d) Use Eq. 5.69, with R \rightarrow \bar{r}, \sigma \rightarrow \rho d \bar{r} , and integrate:

A (r, \theta, \phi)= \begin{cases}\frac{\mu_{0} R \omega \sigma}{3} r \sin \theta \hat{\phi}, & (r \leq R) \\ \frac{\mu_{0} R^{4} \omega \sigma}{3} \frac{\sin \theta}{r^{2}} \hat{\phi}, & (r \geq R)\end{cases}                      (5.69)

A =\frac{\mu_{0} \omega \rho}{3} \frac{\sin \theta}{r^{2}} \hat{\phi} \int_{0}^{R} \bar{r}^{4} d \bar{r}=\frac{\mu_{0} \omega}{3} \frac{3 Q}{4 \pi R^{3}} \frac{\sin \theta}{r^{2}} \frac{R^{5}}{5} \hat{\phi}=\frac{\mu_{0}}{4 \pi} \frac{Q \omega R^{2}}{5} \frac{\sin \theta}{r^{2}} \hat{\phi} .
This is identical to (c); evidently the field is pure dipole, for points outside the sphere.

(e) According to Prob. 5.30, the field is B =\frac{\mu_{0} \omega Q}{4 \pi R}\left[\left(1-\frac{3 r^{2}}{5 R^{2}}\right) \cos \theta \hat{ r }-\left(1-\frac{6 r^{2}}{5 R^{2}}\right) \sin \theta \hat{ \theta }\right] .
The average obviously points in the z direction, so take the z component of \hat{ r }(\cos \theta) \text { and } \hat{ \theta }(-\sin \theta) :

B_{ ave }=\frac{\mu_{0} \omega Q}{4 \pi R} \frac{1}{(4 / 3) \pi R^{3}} \int\left[\left(1-\frac{3 r^{2}}{5 R^{2}}\right) \cos ^{2} \theta+\left(1-\frac{6 r^{2}}{5 R^{2}}\right) \sin ^{2} \theta\right] r^{2} \sin \theta d r d \theta d \phi

 

=\frac{3 \mu_{0} \omega Q}{\left(4 \pi R^{2}\right)^{2}} 2 \pi \int_{0}^{\pi}\left[\left(\frac{r^{3}}{3}-\frac{3}{5} \frac{R^{5}}{5 R^{2}}\right) \cos ^{2} \theta+\left(\frac{R^{3}}{3}-\frac{6}{5} \frac{R^{5}}{5 R^{2}}\right) \sin ^{2} \theta\right] \sin \theta d \theta

 

=\frac{3 \mu_{0} \omega Q}{8 \pi R^{4}} R^{3} \int_{0}^{\pi}\left(\frac{16}{75} \cos ^{2} \theta+\frac{7}{75} \sin ^{2} \theta\right) \sin \theta d \theta=\frac{3 \mu_{0} \omega Q}{8 \pi R} \frac{1}{75} \int_{0}^{\pi}\left(7+9 \cos ^{2} \theta\right) \sin \theta d \theta

 

=\left.\frac{\mu_{0} \omega Q}{200 \pi R}\left(-7 \cos \theta-3 \cos ^{3} \theta\right)\right|_{0} ^{\pi}=\frac{\mu_{0} \omega Q}{200 \pi R}(20)=\frac{\mu_{0} \omega Q}{10 \pi R}(\text { same as (b) }) .

Related Answered Questions

Prove the following uniqueness theorem: If the current density J is specified throughout a volume ν, and either the potential A or the magnetic field B is specified on the surface S bounding ν, then the magnetic field itself is uniquely determined throughout ν. [Hint: First use the divergence

Verified Answer:
Apply the divergence theorem to the function [U × ...

A magnetic dipole m = −m0 zˆ is situated at the origin, in an otherwise uniform magnetic field B = B0 zˆ. Show that there exists a spherical surface, centered at the origin, through which no magnetic field lines pass. Find the radius of this sphere, and sketch the field lines, inside and out.

Verified Answer:
From Eq. 5.88, B _{ tot }=B_{0} \hat{ z }-\...

Using Eq. 5.88, calculate the average magnetic field of a dipole over a sphere of radius R centered at the origin. Do the angular integrals first. Compare your answer with the general theorem in Prob. 5.59. Explain the discrepancy, and indicate how Eq. 5.89 can be corrected to resolve the

Verified Answer:
The issue (and the integral) is identical to the o...

(a) Prove that the average magnetic field, over a sphere of radius R, due to steady currents inside the sphere, is B ave =μ0/4π 2m/R^3,where m is the total dipole moment of the sphere. Contrast the electrostatic result, Eq. 3.105. [This is tough, so I’ll give you a start

Verified Answer:
(a)  B _{ ave }=\frac{1}{(3 / 4) \pi R^{3}}...

A thin glass rod of radius R and length L carries a uniform surface charge σ. It is set spinning about its axis, at an angular velocity ω. Find the magnetic field at a distance s ≫ R from the axis, in the xy plane (Fig. 5.66). [Hint: treat it as a stack of magnetic dipoles.] [Answer:

Verified Answer:
For a dipole at the origin and a field point in th...

A thin uniform donut, carrying charge Q and mass M, rotates about its axis as shown in Fig. 5.64. (a) Find the ratio of its magnetic dipole moment to its angular momentum. This is called the gyromagnetic ratio (or magnetomechanical ratio). (b) What is the gyromagnetic ratio for a uniform spi

Verified Answer:
\text { (a) } I=\frac{Q}{(2 \pi / \omega)}=...

(a) Construct the scalar potential U(r) for a “pure” magnetic dipole m. (b) Construct a scalar potential for the spinning spherical shell (Ex. 5.11). [Hint: for r > R this is a pure dipole field, as you can see by comparing Eqs. 5.69 and 5.87.] (c) Try doing the same for the interior of a solid

Verified Answer:
(a) Exploit the analogy with the electrical case: ...

Just as ∇ · B = 0 allows us to express B as the curl of a vector potential (B = ∇ × A), so ∇ · A = 0 permits us to write A itself as the curl of a “higher” potential: A = ∇ × W. (And this hierarchy can be extended ad infinitum.) (a) Find the general formula for W (as an integral over B), which

Verified Answer:
\text { (a) } \nabla \cdot B =0, \nabla \ti...

Use the Biot-Savart law (most conveniently in the form of Eq. 5.42 appropriate to surface currents) to find the field inside and outside an infinitely long solenoid of radius R, with n turns per unit length, carrying a steady current I.

Verified Answer:
Put the field point on the x axis, so r = (s, 0, 0...

The magnetic field on the axis of a circular current loop (Eq. 5.41) is far from uniform (it falls off sharply with increasing z). You can produce a more nearly uniform field by using two such loops a distance d apart (Fig. 5.59). (a) Find the field (B) as a function of z, and show that

Verified Answer:
(a) From Eq. 5.41, B =\frac{\mu_{0} I R^{2}...