Download Theoretical 1: Magnetic Monopole

Problem Creator: Viktor Ivanov
Theoretical 1: Magnetic Monopole
1
Introduction
Our everyday experience shows that the magnetic poles always exist in pairs North and South.
Breaking a magnet results in the appearance of a new pair of opposite magnetic poles on the
two broken ends. The fundamental laws of physics, however, do not contradict the existence of
magnetically charged particles called magnetic monopoles. The magnetic monopole is an object
possessing just one magnetic pole, either North, or South, which are analogues of the positive
and negative electric charges respectively. Thus, the magnetic field of the monopole is similar
to the electric field created by a static electric charge, i.e. its force lines begin or end at the
point where the monopole is located. This property is in contrast to the closed force lines of the
magnetic field created by permanent magnets (magnetic dipoles) and electric currents. The concept
of magnetic monopole was introduced in 1932 by the famous physicist Paul Dirac. On the basis of
quantum mechanics he proved that the existence of magnetic monopoles can explain the existence
of the elementary electric charge. That is why the physicists do not cease their efforts to discover
magnetic monopoles experimentally.
In the following questions you are going to establish some properties of the magnetic monopoles
by analyzing simple model situations (though experiments). You may assume that all laws of
physics known to you apply to the magnetic monopoles, except the statement for closed force lines
of the magnetic field. The velocities considered in this problem are much smaller than the speed
of light and, therefore, you may neglect the relativistic effects on time, length and mass.
Use the following physical constants in your solution:
2
magnetic permeability of vacuum:
µ0 = 4π × 10−7 H/m ;
electric permittivity of vacuum:
0 = 8.85 × 10−12 F/m ;
speed of light:
c = 2.998 × 108 m/s ;
elementary electric charge:
e = 1.602 × 10−19 C ;
Planks constant:
h = 6.626 × 10−34 J.s .
Questions
1. When exposed to external magnetic field of induction B, a monopole of magnetic charge qm
experiences a force:
F = qm B
(1)
(a) Derive the unit of magnetic charge in terms of the basic SI units: kilogram, meter, second,
ampere.
(0.8 points)
Magnetic Monopole
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Questions
Question 1. When exposed to external magnetic field of induction B, a mon
Problem Creator: Viktor Ivanov
charge qm experiences a force:
Questions
osed to external magnetic field of induction B, a monopole of magnetic
F qm B
a force:
Theoretical 1: Magnetic Monopole
F
a. Derive the unit of magnetic charge in terms of the basic SI units: kilogra
ampere.
qm B
2. Electric current I circulates along a circular loop of radius a. A monopole of magnetic charge
of magnetic charge
terms ofonthe
SItheunits:
kilogram,
2.a Electric
currentsecond,
I circulates
a circular
loop of radius a
qm in
is situated
thebasic
axis ofQuestion
loop at
point ofmeter,
coordinate
z relative toalong
its center,
as shown
qm Zis and
situated
on the axis
of thecirculation
loop at a are
point of coordina
in Figure 1 . The positive magnetic
direction ofcharge
the axis
the direction
of current
related through the right-hand
rule.as shown in Figure 1. The positive direction of the axis Z and the di
center,
current I circulates
a an
circular
loop
a. AFthrough
monopole
of
circulation
are related
right-hand
(a) along
Find out
expression
for of
the radius
z-component
force
acting onrule.
the monopole.
z of the the
(1.1 points)
situated on the axis of the loop at a point of coordinate z relative to its
a. Find out an expression for the z-component Fz of the force acting on the
gure 1. The positive direction of the axis Z and the direction of current
I
hrough the right-hand rule.
a
qm
ression for the z-component Fz of the force acting on the monopole.
Z
qm
Figure 1.
Figure 1: Question 2
Z
Question 3. When at rest, the magnetic monopole creates static magnetic fi
3. When at rest, the magneticelectric
monopole
creates
staticby
magnetic
field, similar
to the
fieldinduction B at a
field
produced
a static electric
charge.
Theelectric
magnetic
produced by a static electric charge. The magnetic induction B at a point of position- vector
vector r, relative to the monopole (see Figure 2), is given by the equation:
r, relative to the monopole (see Figure 2), is given by the equation:
km qm r
k m qm r
B=
,
rest, the magnetic monopole creates static magnetic
field,r3 similar
to theB
r3
by a static electric charge. The magnetic induction B at a point of positionwhere km is a coefficient of proportionality and r = |r|.
km is a coefficient of proportionality and r
monopole (see Figure 2), is given by thewhere
equation:
B
(2)
r.
qm
Figure 2.
r
a. By analyzing theB system described in Question 2, express the unkno
throughrthe provided fundamental constants.
k m qm r
r3
qm
t of proportionality and r
r.
b. Formulate by means of equation the Gauss law for the flux
Figure
2.
created by the magnetic monopole.
Figure 2: Question 3
of the m
Question
A moving
electric charge
he system described in Question 2, express
the 4.
unknown
coefficient
km creates magnetic field. Likewise, the
(a) By analyzing the system described in Question 2, express the unknown coefficient km
monopole produces electric field with circular force-lines
(i.e. a vortex field) c
vided fundamental constants.
through the provided fundamental constants.
(1.6 points)
direction of motion of the monopole (see Figure 3). Consider a monopole of m
(b) Formulate by means of equation the Gauss law for the flux Φ of the magnetic induction
. points)
along
lineinduction
with a constant velocity v(0.5
equation thecreated
Gaussby
law
the moving
flux
of thea straight
magnetic
thefor
magnetic
monopole.
means of
magnetic monopole.
4. A moving electric charge creates magnetic field. Likewise, the moving magnetic
monopole
4
produces electric field with circular force-lines (i.e. a vortex field) concentric with the direction
g electric charge ofcreates
magnetic field. Likewise, the moving magnetic
motion of the monopole (see Figure 3). Consider a monopole of magnetic charge qm moving
ctric field with circular
force-lines
(i.e. aaconstant
vortex field)
with the
along a straight
line with
velocityconcentric
v.
the monopole (see Figure 3). Consider a monopole of magnetic charge qm
line with a constant velocity v.
Magnetic Monopole
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Problem Creator: Viktor Ivanov
Theoretical 1: Magnetic Monopole
r
qm
electric force line
v
Figure 3.
Figure 3: Question 4
an expression
for electric
the intensity
of the electric
created in
byathe monopole in a
(a) Derive an expression a.
for Derive
the intensity
E of the
fieldEcreated
by thefield
monopole
point of position-vector rpoint
making
an angle θ with
the vector
velocity,
shown
in of
Figure
r making
an of
angle
withasthe
vector
velocity, as shown in
of position-vector
3. Use vector notations in
your 3.
final
order to in
specify
the magnitude,
Figure
Useanswer
vectorinnotations
your both,
final answer
in order and
to specify both, the
the direction of the electric
field.
(1.7
points)
magnitude, and the direction of the electric field.
(b) Suppose that a positive electric charge qe and a positive magnetic charge qm are movb. Suppose that a positive electric charge qe and a positive magnetic charge qm are moving
ing toward you, perpendicularly to the sheet of paper. Draw
an arbitrary force line
against you, perpendicularly to the sheet of paper. Draw two arbitrary force lines in you
in your answer sheet for the magnetic field created by the electric charge. Draw sepaanswer sheet – one for the magnetic field created by the electric charge and another, fo
rately another arbitrary force line for the electric field created by the magnetic charge.
field created by the magnetic monopole. Indicate
directions of the two
Indicate the directions ofthe
theelectric
two lines.
(0.4 the
points)
lines.
5. The analogy between electric and magnetic charges is found also in the way they interact with
5. Therespectively.
analogy between
electrictoand
is found
external magnetic and Question
electric fields
Similarly
themagnetic
Lorentz charges
force acting
on also
an in the way they
with field,
external
electric
fields respectively.
Similarly
to the Lorentz force
electric charge moving interact
in magnetic
themagnetic
magneticand
charge
experiences
a force when
it moves
acting
on
an
electric
charge
moving
in
magnetic
field,
the
magnetic
charge
experiences a force
in electric field.
when it moves in electric field.
(a) Propose and analyze a thought experiment in order to derive an expression for the Lorentz
force acting on a monopole
of magnetic
charge
qm moving
with ain velocity
in electric
a. Propose
and analyze
a thought
experiment
order tovderive
an expression for the
field of intensity E. Use vector
notations
in
your
final
answer
in
order
to
specify
both, the
with a velocity v in
“Lorenz” force acting on a monopole of magnetic charge qm moving
magnitude and the direction
of
the
force.
When
describing
your
thought
experiment,
use
electric field of intensity E. Use vector notations in your final answer in order to specify
proper drawings and short
comments
to themand
instead
of a lengthy
(1.0 points)
both,
the magnitude
the direction
of thetext.
force. When
describing your though
experiment,
use propertodrawings
and short
comments
to them
instead of a lengthy text.
6. A point particle of electric charge
qe is confined
move along
a circle
without
any resistance
or friction, as shown inQuestion
Figure 4.6. A
of magnetic
charge qqmis passes
through
plane
A monopole
point particle
of electric charge
confined
to movethe
along
a circle without any
e
of the circle by movingresistance
along itsoraxis
Z
from
z
→
−∞
to
z
→
∞.
friction, as shown in Figure 4. A monopole of magnetic charge q passes through
m
the plane of the circle by moving along its axis Z from z
to z
.
qe
qm
Z
Figure 4.
Figure 4: Question 6
5
Magnetic Monopole
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a. Obtain an expression for the change in Z–component, Lz, of the angular momentum of
Problem Creator:
Viktor Ivanov
the electrically
charged particle during the whole motion of the magnetic monopole.
Express your answer in terms of qe, qm and fundamental constants only.
Theoretical
1:monopoles
Magnetic
Monopole
Question 7. In his famous
work on magnetic
Paul Dirac
has argued that if just one
single magnetic monopole existed in the Universe, all electric charges should be multiple of a
specific elementary electric charge, whose magnitude is related to the magnetic charge of that
(a) Obtain
an expression
for itthe
change
Zcomponent,
∆Lz , ofwhich
the angular
momentum
monopole.
Historically,
is the
firstinhypothesis
in physics,
explains
the existence of the
of
the
electrically
charged
particle
during
the
whole
motion
of
the
magnetic
monopole.
elementary electric charge.
Express your answer in terms of qe , qm and fundamental constants only.
(1.3 points)
the system
described
in Question
6, assuming
that that
all magnetic
monopoles
7. In hisConsider
famous work
on magnetic
monopoles
Paul Dirac
has argued
if just one
magnetic existing in
qm respectively
.
the Nature
magnetic
of the
sameshould
magnitude,
+qm and
monopole
existedhave
in the
Universe,charges
all electric
charges
be multiple
of a–specific
elementary
electric charge, whose magnitude is related to the magnetic charge of that monopole. Historithein concepts
of quantum
physics
to the
motion
of electrically
charged
cally, it isa.theBy
firstapplying
hypothesis
physics, which
explains the
existence
of the
elementary
electric
particle along the circular orbit, derive a relationship between the elementary electric
charge.
, assumedintoQuestion
be the charge
of thethat
electron,
and the
magneticexisting
charge qm of the
Consider thecharge
system edescribed
6, assuming
all magnetic
monopoles
in the Naturemonopole.
have magnetic
charges
the same magnitude, +qm and −qm respectively.
qm of
numerically.
Calculate
(a) By applying the concepts of quantum physics to the motion of electrically charged
particle
2
b. the
The
electron
possesses
self magnetic
moment
pm 9.274electric
10 24 A.m
along
circular
orbit,
derive aa relationship
between
theofelementary
charge. By
e, assuming
assumedthat
to be
charge of
the electron,
and
the magnetic
qm of
monopole.
thethe
magnetic
properties
of the
electron
are duecharge
to a pair
of the
spatially
separated point
Calculate qm numerically.
(1.1 points)
magnetic monopoles of opposite magnetic charges, +qm and –qm respectively; calculate the
(b) The electron
possesses
a self these
magnetic
moment of pm = 9.274 × 10−24 A.m2 . By assuming
distance
d between
monopoles.
that the magnetic properties of the electron are due to a pair of spatially separated point
magnetic monopoles of opposite magnetic charges, +qm and −qm respectively; calculate
the distance d between these monopoles.
(0.5 points)
Useful math:
Useful math :
The solid
angle byenclosed
by half-opening
a cone of half-opening
angle
(see the
• The solid angle
Ω enclosed
a cone of
angle θ (see
the Figure
5) figure)
is: Ω =is:
2π(1 − cos(θ))
2 (1 cos( ))
Figure 5: Solid Angle
Depending on your approach to the solution you may need the following integral:
• Depending on your approach to the solution you may need the following integral:
Z ∞
dz
2
dz
2
= 2.
3/2
2
2
2
2 32
2
a
−∞ (z + a )
(z a )
a
(3)
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Magnetic Monopole
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