Indiana University Physics P332: Electromagnetism Homework #4 (Due Friday 2/15/2019)

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Percival Hodge
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Indiana University Physics P332: Electromagnetism Homework #4 (Due Friday 2/15/2019) Reading: Griffiths Chapter 6 1. Conceptual Problems 2. Griffiths 6.6 3. Griffiths 6.12 4. Griffiths 6.13 5.

1 Indiana University Physics P332: Electromagnetism Homework #4 (Due Friday 2/15/2019) Reading: Griffiths Chapter 6 1. Conceptual Problems 2. Griffiths Griffiths Griffiths Griffiths Consider a toroid with a soft iron core as shown in the figure. The toroid is essentially a solid iron cylinder of radius a and length 2πs, bent into a ring, and then wrapped with N turns of a wire through which a current I is flowing. (a) Compute H at a point in the interior of the torus, at a radius s as shown, as a function of the current I. What is the direction of H? (b) For small enough currents, the magnetization M will be far from saturation and is approximately linearly proportional to H. In this regime, suppose the iron has a relative permeability µ/µ 0 = 200. What is the magnetic field B at a point at radius s? What is the direction of B? (c) Are there any bound currents? If so, what are they? (d) Suppose that you excise a thin disc (of thickness d where d a) of iron from the toroid, such that that there is now a gap in the iron core. Consider the case where also a s. What is the magnetic field in the center of the gap? Evaluate this for the case d/a = 0.1. (Hint: Use superposition!) 1

2 Amperian Loop s I 7. This problem explores how we know that the magnetic dipole moment of the proton (and the field it produces) is due to circulating currents of its quark constituents and not due to a suitable distribution of real magnetic charge. (a) Draw the magnetic field lines in the two cases, showing that while the dipole field outside the proton would look the same, the field inside the proton would be different. Consider the magnetic field inside the proton as being due to a magnetization current density K b = M ˆn in one case, and the surface charge density σ b = M ˆn in the other case, where M is uniform magnetization density of the proton, m p = (4πRp/3)M. 3 (b) One approach to make a choice between these two scenarios is to probe the internal magnetic structure of the proton. The hyperfine splitting of the hydrogen atom occurs when the magnetic dipole moment of the electron m e interacts with the dipole magnetic field of the proton B p ( r). Recall that the potential energy of a dipole in an external magnetic field is U = m e B p ( r). Given spherical symmetry of the 1s electron wave function (note: you do not need to know this wave function, only that it is spherically symmetric!), show that there is no effect due to the exterior field of the proton. (c) On the other hand, the 1s wave function has a nonzero amplitude at the position r 0 of the proton. Show that in each case the magnetic field of the proton can be written as B p current ( r) = αµ 0 m p δ( r r 0 ) and B p charge ( r) = βµ 0 m p δ( r r 0 ), where α and β are constants and m p is the magnetic dipole moment of the proton. Find α and β, showing that they are different. Therefore the hyperfine shift measured from high-precision experiments can be used to determine which of these models 2

3 is correct. 8. In this problem you will explore the behavior of real electromagnet: (a) A cylindrical solenoid occupies the interval L/2 z L/2, has radius R and is wound tightly with N turns of a wire which carries current I. Use superposition and derive an expression for the magnetic field on the symmetry axis in the form (z) = (µ 0 NI/L) f(z)ẑ. Compute the ratio B(±L/2)/ (0) in the limit when L R. (b) Fill the solenoid with a close-fitting rod of soft iron with effective susceptibility χ m. Make the approximation M( r) = M(z)ẑ. Find the amplification of the magnetic field by the soft iron at z = 0 and z = ±L/2, namely B(0)/ (0) and B(L/2)/ (L/2) and comment on the result when χ m 1. 3

A"very"long"aluminum"(paramagne1c!)"rod" carries"a"uniformly"distributed"current"i"along" the"+z"direc1on."we"know"b"will"be"ccw"as" viewed"from"above."(right?

4 A"very"long"aluminum"(paramagne1c!)"rod" carries"a"uniformly"distributed"current"i"along" the"+z"direc1on."we"know"b"will"be"ccw"as" viewed"from"above."(right?)" "What"about"H#and"M#inside"the"cylinder?" A) Both are CCW B) Both are CW C) H is CCW, but M is CW D) H is CW, M is CCW E)??? A"very"long"aluminum"(paramagne1c!)"rod" carries"a"uniformly"distributed"current"i"along" the"+z"direc1on."what"is"the"direc1on"of"the" bound"volume"current?" A) J B points parallel to I B) J B points anti-parallel to I C) It s zero! D) Other/not sure A"very"long"aluminum"(paramagne1c!)"rod" carries"a"uniformly"distributed"current"i"along" the"+z"direc1on."" What"is"the"direc1on"of"the"bound"surface# current?" What"if"that"long"rod"(the"wire)"was"made"of" copper"(diamagne1c!)"instead.""of"b,"m,"h," and"j_bound,"which"ones" flip"sign?"" A) K B points parallel to I B) K B points anti-parallel to I C) Other/not sure A) All 4 flip B) 3 of the 4 flip C) 2 of the 4 flip D) 1 of them flips E) None of them flips I The para case The dia case

What"if"that"long"rod"(the"wire)"was"made"of" copper"(diamagne1c!)"instead.""of"b,"m,"h," and"j_bound,"which"ones" flip"sign?"" Inside a hollow solenoid, B= =µ 0 ni. What is the formula for H inside?

5 What"if"that"long"rod"(the"wire)"was"made"of" copper"(diamagne1c!)"instead.""of"b,"m,"h," and"j_bound,"which"ones" flip"sign?"" Inside a hollow solenoid, B= =µ 0 ni. What is the formula for H inside? A) All 4 flip B) 3 of the 4 flip C) 2 of the 4 flip D) 1 of them flips E) None of them flips I B M H The para case The dia case Inside a hollow solenoid, B= =µ 0 ni, ( so H=H 0 =ni ) If the solenoid is filled with a normal paramagnetic material, like aluminum, what is B inside?... Inside a hollow solenoid, B= =µ 0 ni, ( so H=H 0 =ni ) If the solenoid is filled with iron, what is H inside?... A) Still exactly B) a little more than C) a lot more D) a little less than E) a lot less than A) H 0 B) a little more than H 0 C) a lot more D) a little less than H 0 E) a lot less than H 0

6 A"very"long"rod"carries"a"uniformly"distributed" current"i"along"the"+z"direc1on."" Compare"the"BOfield"OUTSIDE"when"the"rod"is" a"paramagnet"(e.g."al)"to"the"bofield"outside" when"the"rod"is"a"diamagnet"(e.g."cu)" " B outside the paramagnetic rod is A) Slightly smaller than B) The same as C) Slightly larger than has a uniform field throughout its bulk, and thus a uniform H 0 =?? B outside the diamagnetic rod has a uniform field throughout its bulk, and thus a uniform H 0 = /µ = / µ 0 (1+χ M ) We then cut out a cylindrical hole (very skinny, very tall!) has a uniform field B0 throughout its interior. We cut out a cylindrical hole (very skinny, very tall!) What is M at the center of that hole? Α) χ Μ H 0 B) little more than χ Μ H 0 C) Little less than χ Μ H 0 D) Zero E)??? (it depends/not sure) What is B at the center of that hole? A) B) more than C) less than D)??

7 has a uniform field B0 throughout its interior. We cut out a wafer-like hole (very wide, very short!) What is B at the center of that hole? A) B) more than C) less than D)??

MAGNETIZATION, BOUND CURRENTS AND H

MAGNETIZATION, BOUND CURRENTS AND H 6.4 A solid cylinder has uniform magnetization M throughout the volume in the x direction as shown. What's the magnitude of the total magnetic dipole moment of the cylinder?