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Chapter 5: Q57P (page 262)

A magnetic dipole m⇀=m0z^ is situated at the origin, in an otherwiseuniform magnetic field B⇀=B0z^ . 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.

Short Answer

Expert verified

There exists a spherical surface of radius μ0m02πB03, centered at the origin, through which no magnetic field lines pass.

Step by step solution

01

Given data

There is an magnetic dipole m⇀=-m0z^ is situated at the origin, in an uniform magnetic field B⇀=B0z^.

02

Magnetic field due to a dipole

The magnetic field due to a magnetic dipole m⇀ is

localid="1658559878707" Bdip=μ0m4π3r[2cosθr^+sinθθ^]

Here, μ0 is the permeability of free space.

03

Net magnetic field near origin

From equation (1), the net magnetic field near the origin is,

B⇀=B0z^+Bdip→=B0z^-μ0m04πr32c0sθr^+sinθθ^

The radial component of this field is,

B⇀.r^=B0cosθ-μ0m04πr32cosθ=B0-μ0m02πr3cosθ

Thus, the net field is zero for any value of θ at radius R where

localid="1657777042303" B0-μ0m02πR3=0R=μ0m02πB33

The field lines are shown in the following figure


Thus, the magnetic field lines are absent in the sphere of radiusμ0m02πB33 .

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Most popular questions from this chapter

(a) one way to fill in the "missing link" in Fig. 5.48 is to exploit the analogy between the defining equations for A(viz∇-A=0,∇×A=B)and Maxwell's equations forB(viz∇.B=0∇×B=μ0J).Evidently A depends on B in exactly the same way that B depends onμ0J(to wit: the Biot-Savart law). Use this observation to write down the formula for A in terms of B.

(b) The electrical analog to your result in (a) is

localid="1658557463395" V(r)=-14π∫E(r')-r^r2dτ'

Derive it, by exploiting the appropriate analogy.

A circular loop of wire, with radius , R lies in the xy plane (centered at the origin) and carries a current running counterclockwise as viewed from the positive z axis.

(a) What is its magnetic dipole moment?

(b) What is the (approximate) magnetic field at points far from the origin?

(c) Show that, for points on the z axis, your answer is consistent with the exact field (Ex. 5.6), when z>>R.

Find and sketch the trajectory of the particle in Ex. 5.2, if it starts at

the origin with velocity

(a)v→(0)=EBy⏜(b)v→(0)=E2By⏜(c)v→(0)=EB(y⏜+z⏜).

Consider the motion of a particle with mass m and electric charge qein the field of a (hypothetical) stationary magnetic monopole qmat the origin:

B=μ04qmr2r^

(a) Find the acceleration of qe, expressing your answer in terms of localid="1657533955352" q, qm, m, r (the position of the particle), and v(its velocity).

(b) Show that the speed v=|v|is a constant of the motion.

(c) Show that the vector quantity

Q=m(r×v)-μ0qeqm4πr^

is a constant of the motion. [Hint: differentiate it with respect to time, and prove-using the equation of motion from (a)-that the derivative is zero.]

(d) Choosing spherical coordinates localid="1657534066650" (r,θ,ϕ), with the polar (z) axis along Q,

(i) calculate , localid="1657533121591" Q‐ϕ^and show that θis a constant of the motion (so qemoves on the surface of a cone-something Poincare first discovered in 1896)24;

(ii) calculate Q‐r^, and show that the magnitude of Qis

Q=μ04π|qeqmcosθ|;

(iii) calculate Q‐θ^, show that

dϕdt=kr2,

and determine the constant k .

(e) By expressing v2in spherical coordinates, obtain the equation for the trajectory, in the form

drdϕ=f(r)

(that is: determine the function )f(r)).

(t) Solve this equation for .r(ϕ)

Just as V.B=0allows us to express B as the curl of a vector potential (B=∇×A), so ∇.A=0permits 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 holds whenB→0 at ∞.

(b) Determine for the case of a uniform magnetic field B. [Hint: see Prob. 5.25.]

(c) Find inside and outside an infinite solenoid. [Hint: see Ex. 5.12.]
See all solutions

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